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a/content/numpy-stars.rst b/content/numpy-stars.rst index b9ec907b..d6fecc6c 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -32,18 +32,233 @@ It helps you work with "data", as in large amounts of numbers, by providing: 2. Efficient routines to manipulate data, such as reading and writing, splitting and concatenation, and other transformations 3. Efficient routines to perform computations on data, such as statistics and linear algebra +Let's explore the fundamentals of this foundational library by analyzing some astronomical data. + .. highlight:: python Exploring our local stellar neighbourhood ----------------------------------------- -The Gaia sattelite has imaged around 1.8 billion stars and other objects. -Among the things it measured was the "parallax" of each object, which is how much a star seems to "move" in relationship to other stars as the sattelite traveled around the sun and observed it from different angles. -It's the same thing as how nearby objects appear to be in different positions when you look at them with your left versus your right eye. +The `Gaia sattelite `__ has imaged around 1.8 billion stars and other objects. +Among the things it measured was the "parallax" of each object, which is how much it seems to "move" in relationship to very distant stars as the sattelite traveled around the sun and observed it from different angles. We can use the amount of parallax to determine how far away a star is. .. image:: img/numpy/parsec.png + :width: 300px + :align: center + +A sample of the Gaia data can be found in :download:`../resources/data/numpy/local_stars.csv`. +This is a *comma separated value (CSV)* file, where each line contains 6 numbers describing a single object. +The first line of that file is the "header" indicating the meaning of each number:: + + with open("../resources/data/numpy/local_stars.csv") as f: + # The file handle `f` acts as a generator that yields lines of text. + first_line = next(f) # take a single item from the generator + names = first_line.split(",") # split on commas to get a list of individual names + print(names) + +We can use one of NumPy's functions to load the entire file into memory:: + + import numpy as np # we usually use `np` as shorthand + # We set `skip_header=1` to skip the first line containing the description of the values. + data = np.genfromtxt("../resources/data/numpy/local_stars.csv", delimiter=",", skip_header=1) + print(data) + +The ``data`` variable is a NumPy array:: + + print(type(data)) + What is an array? ----------------- +An array is a "grid" of values, with all the same type, tightly packed into memory side-by-side. +This is why we skipped the first line of the file containing ``str`` type descriptions, so that we were only reading numbers. +In our case, the data type of the array is:: + + print(data.dtype) + +Arrays can have any number of dimensions:: + + print(data.ndim) + +You can think of a 1D array as a *list* and a 2D array as a *table*. +In mathematics, we call a 1D array a *vector*, a 2D array a *matrix*, and an array with 3 or even more dimensions a *tensor*. +An array can even have zero dimensions if it represents a single *scalar* number. +Regardless of how many dimensions the array has, it is always a Python object of type :class:`numpy.ndarray`. + +Let's see how many stars we have:: + + print(data.shape) + +We have information on more than a million stars, and for each star we have 6 values, corresponding to (see ``names`` above): + +0. ``right_ascention``: horizontal position of the star along the celestial equator (degrees) +#. ``declination``: vertical position of the star above/below celestial equator (degrees) +#. ``parallax``: amount the star's position in the sky changes throughout the year +#. ``magnitude_g``: how bright the star appears in our sky (inverse Logarithmic scale) +#. ``magnitude_red``: red-component of the color of the star +#. ``magnitude_blue``: blue-component of the color of the star + + +Working with arrays +------------------- + +The way the array is laid out in memory makes it very fast to select portions of the data, by "indexing" or "slicing" the array. +You can index/slice arrays in a similar manner as Python lists, but you specify an index/slice for each dimension. +At the moment, we are only interested in the parallax value of each star, so let's extract that column and store it in a variable of its own:: + + parallax = data[:, 2] # `:` is a shorthand for "select everything along this dimension" + print(parallax) + +Now for some basic math. +The parallax values are currently in `milli-arc-seconds `__. +Let's convert that into distance. +A common distance metric is the `"parsec" `__, which is the distance at which an object has a parallax of exactly one arc-second, which is 3.26156 light-years:: + + distance_parsecs = 1 / (parallax / 1000) # 1000 milli-arc-seconds in an arc-second + distance_ly = distance_parsecs * 3.26156 + +Note that we didn't write any ``for``-loops. +In NumPy, arithmetic operations on arrays are applied to every element in that array. +It is very fast, because it is implemented in C or Fortran and makes use of specialized CPU instructions that perform operations on many values at once. +We say an operation is "vectorized" when the looping over elements is carried out by NumPy internally. +Common mathematical operations include: ``+`` (:data:`numpy.add`), ``-`` (:data:`numpy.subtract`), ``*`` (:data:`numpy.multiply`), ``/`` (:data:`numpy.divide`), ``.T`` (:func:`numpy.transpose`), :data:`numpy.sqrt`, :func:`numpy.sum`, :func:`numpy.mean`, ... + +.. note:: + In NumPy, ``*`` performs element-wise multiplication and ``@`` performs matrix multiplication. This is different from other languages such as MATLAB or Julia. + +Let's examine the closest star in the dataset:: + + distance_to_closest_star = distance_ly.min() # returns the minimum value + index_of_closest_star = distance_ly.argmin() # returns the index of the minimum value + print(distance_to_closest_star, index_of_closest_star) + +The closest star is `Proxima Centauri `__ at 4.2465 light-years. +To see all the information we have on it, we can select the entire row using the index we just obtained:: + + print(names) # the column names for reference + print(data[index_of_closest_star, :]) # select the row corresponding to Proxima Centauri + +NumPy has many other ways to select data. +For example, let's see if we can show some information on the 10 stars closest to us. +To do this, we must first order the stars by distance:: + + order_by_distance = np.argsort(distance_parsecs) # argsort returns indices, not values + closest_indices = order_by_distance[:10] # select the first 10 elements + print(closest_indices) + +We now have a list (technically a 1D numpy.array) of integer indices of the stars closest to us. +We can select multiple rows by indexing the array with such a list:: + + closest_stars = data[closest_indices, :] + print(distance_ly[closest_indices]) + +Another powerful indexing technique is to create an index based on boolean values. +Recall that in Python, comparison operations (``==``, ``<``, ``<=``, ``>``, ``>=``) produce a boolean (``True``/``False``) value. +See what happens when you apply them to an array:: + + closer_than_15_ly = distance_ly < 15 + print(closer_than_15_ly.shape) + print(closer_than_15_ly[:10]) + +It produces a new array, of the same dimensions as the original array, but filled with ``True``/``False`` values indicating for each element whether the comparison holds true or not. +We call this a "boolean mask" and we can use it to index another array:: + + print(distance_ly[closer_than_15_ly]) + print(data[closer_than_15_ly, :].shape) + +Can we find the `Big Dipper `__? +It lies within a section of the sky with a right ascention from 160 to 210 degrees, and a declination from 45 to 65 degrees:: + + right_ascention = data[:, 0] + declination = data[:, 1] + boolean_mask = ( + (right_ascention > 160) & (right_ascention < 210) & + (declination > 45) & (declination < 65) + ) + sky_section = data[boolean_mask] + print(sky_section.shape) + +In our dataset, there are more than 15k stars in our chosen section of sky. +The Big Dipper should consist of the 7 brightest ones. +Column 3, named `magnitude_g` denotes bright the star looks as seen by the Gaia sattelite. +The `magnitude scale `__ works "in reverse", such that very bright stars have a magnitude of 1 and weaker stars have a larger magnitude. + +.. code:: + order_by_magnitude = np.argsort(sky_section[:, 3]) + sky_section = sky_section[order_by_magnitude, :] + big_dipper = sky_section[:7] + + # Make a figure with the position of the stars. We will cover this in a later lesson. + fig, ax = plt.subplots() + ax.scatter(big_dipper[:, 0], big_dipper[:, 1]) + ax.xaxis.set_inverted(True) + ax.set_aspect("equal") + ax.set_xlabel("right ascention (degrees)") + ax.set_ylabel("declination (degrees)") + + +.. seealso:: + + :ref:`Numpy basic indexing docs ` + +Clever and efficient use of these operations is a key to NumPy's speed: you should try to cleverly use these selectors (written in C) to extract data to be used with other NumPy functions written in C or Fortran. +This will give you the benefits of Python with most of the speed of C. + + +Exercises 1 +----------- + +.. challenge:: Exercises: Numpy-1 + + Proxima Centauri is part of a 3-star system called `Alpha Centauri `__. + The other two stars are very close together and appear as a single object (Alpha Centauri AB) in this dataset. + + #. What is the row index of the binary star Alpha Centauri AB? + It is the second closest star to us. + What is the distance from us to Alpha Centauri AB? + #. What is the distance from us to the brightest star in the dataset (`Sirius `__)? + #. How many stars are there in the dataset that are even brighter than the brightest star in the Big Dipper (`Alioth `__)? + + +Creating arrays +--------------- + +Now we know how far stars are from us, but how far are stars from each other? +This is easier to answer if we transform the coordinates from Gaia-CRF (ra, dec, distance) in degrees -> Cartesian (x, y, z) in light-years. +Let's create a new 2D array of shape ``(n_stars, 3)`` with the Cartesian coordinates of each star, measured in light-years, with us at the center. + +We can compute each coordinate using the trigonometry functions `np.sin` and `np.cos`. +These operate in radians, so we use `np.deg2rad()` to convert degrees to radians:: + + x = distance_ly * np.cos(np.deg2rad(declination)) * np.cos(np.deg2rad(right_ascention)) + y = distance_ly * np.cos(np.deg2rad(declination)) * np.sin(np.deg2rad(right_ascention)) + z = distance_ly * np.sin(np.deg2rad(declination)) + +One of the many ways to create an array is to start with a Python list. +Now currently have three 1D arrays. +To create a 2D array out of them, we packing them into a Python list and then convert this list to an array:: + + xyz_list = [x, y, z] + xyz = np.array(xyz_list) + print(xyz.shape) + +This gives us an array with 3 rows and each star being a column. +To swap rows and columns is called the `Transpose `__ and is such a common operation, especially when doing linear algebra, that NumPy has a shorthand for it:: + + xyz = xyz.T # transpose the 2D array + print(xyz.shape) + +# Question: Alkaid and Mizar form the tail end of the "big dipper". +# How far are these stars from one another? +# * Dubhe +# * Mizar Megrez +# / * * +# * Alkaid / Alioth * Merak +# \ / * +# \ / Phecda +# \ / +# \ / +# * Sol +# From 18277a0bbec92b44cd2f2e7125ad71895c0ab055 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Mon, 29 Jun 2026 23:41:32 +0300 Subject: [PATCH 03/31] Continue working on NumPy lesson --- content/img/numpy/Big_Dipper.jpg | Bin 0 -> 267280 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Image `retrieved from WikiMedia `__. CC BY license. +**Left:** The Big Dipper as seen from Fujian. Image `retrieved from WikiMedia `__. **Right:** The Big Dipper on a sky chart. Image `retrieved from in-the-sky.org `__. Both images have a CC-BY license. It lies within a section of the sky with a `right ascension `__ from 160 to 210 degrees, and a `declination `__ from 45 to 65 degrees:: From 1f50f77fa269416d05551bd714a62561beb98cf4 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Tue, 13 Oct 2026 09:51:34 +0300 Subject: [PATCH 22/31] Better Hertzsprung-Russel diagram --- content/numpy-stars.rst | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index df2bc52f..793f2b22 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -302,12 +302,13 @@ We can compute this by taking the difference between the blue and red components Now we have everything we need to make the plot. Again, just take this plotting code at face value for now:: import matplotlib.pyplot as plt + from matplotlib.colors import LinearSegmentedColormap + plt.style.use("dark_background") plt.figure(figsize=(6, 8)) - plt.scatter( - color, magnitude, s=1, c=distance_ly, cmap="plasma", alpha=0.1 + cmap = LinearSegmentedColormap.from_list( + "stars", ["#ccfeff", "#ffffff", "#fff18f", "#e5d000", "#dd6a00", "#cc0000", "#6b0000"] ) - cb = plt.colorbar(label="Distance (light years)", fraction=0.05, aspect=40) - cb.solids.set(alpha=1) # make the colors in the colorbar not be transparent + plt.scatter(color, magnitude, s=1, c=color, cmap=cmap, alpha=0.1, clim=(0, 3)) plt.gca().invert_yaxis() # put bright stars on top, dim stars at the bottom plt.title("Hertzsprung-Russell diagram") plt.xlabel("Color (Blue ⇨ Red)") From bd80624ad64be0e4e77b5ff1c9aba17737f4d2f7 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Tue, 13 Oct 2026 13:48:59 +0300 Subject: [PATCH 23/31] small tweaks --- content/numpy-stars.rst | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index 7cabbf0f..289fa2a0 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -99,7 +99,7 @@ Let's see how many stars we have. The ``.shape`` property of an array contains t print(data.shape) -We have information on more than a million stars, and for each star we have 6 values, corresponding to 6 columns listed above. +We have information on more than a million stars, and for each star we have 6 values. Working with arrays @@ -172,10 +172,10 @@ To do this, we must first order the stars by distance:: print(closest_indices) We now have a list (technically a 1D :class:`numpy.ndarray`) of integer indices of the stars closest to us. -We can select multiple rows by indexing the array with such a list:: +We can select multiple elements by indexing the array with such a list:: - closest_stars = data[closest_indices, :] - print(distance_ly[closest_indices]) + print(data[closest_indices, :]) # selecting multiple rows in a 2D array + print(distance_ly[closest_indices]) # selecting multiple elements in a 1D array Can we find the `Big Dipper `__? From ef876e3b78bfb6099e923829fd60906d2b6c54e3 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Thu, 15 Oct 2026 10:33:25 +0300 Subject: [PATCH 24/31] simplifying lesson --- content/numpy-stars.rst | 168 ++++++++++++++-------------------------- 1 file changed, 57 insertions(+), 111 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index 7e733a08..825497a4 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -55,15 +55,15 @@ We can use the amount of parallax to determine how far away a star is. Visual explanation of the parallax effect. Image `retrieved from WikiMedia `__. Public domain. A sample of the Gaia data can be found in :download:`../resources/data/numpy/local_stars.csv`. -This is a *comma separated value (CSV)* file, where each line contains 6 numbers describing a single object. -The first line of that file is the "header" indicating the meaning of each number:: - - right_ascension,declination,parallax,magnitude_g,magnitude_blue,magnitude_red - 0.0001786275254186,8.371651684580781,6.857487064725919,12.039392,12.56381,11.3628235 - 0.0003111955790572,58.5575410957825,2.5185016849859228,12.374151,12.819315,11.755815 - 0.0005898233877197,5.380517264865798,6.660454655312786,12.95553,13.5889015,12.193393 - 0.0010609752955102,-55.64880544379682,7.89357866196891,13.52746,14.636247,12.455662 - 0.0011307510147073,3.0206027981262924,4.043843423571794,12.745788,13.150434,12.17218 +This is a *comma separated value (CSV)* file, where each line contains the parallax, magnitude and color of the star. +The first line of that file is the "header" containing the column names:: + + parallax,magnitude,color + 0.006857487064725919,12.039392,1.2009865000000008 + 0.002518501684985923,12.374151,1.0634999999999994 + 0.0066604546553127855,12.95553,1.3955085 + 0.00789357866196891,13.52746,2.180584999999999 + 0.004043843423571794,12.745788,0.9782539999999997 ... We can use one of NumPy's functions to load the entire file into memory:: @@ -107,40 +107,30 @@ Working with arrays Here is the data, formatted as a table: -=============== =========== ======== =========== ============== ============= -right_ascension declination parallax magnitude_g magnitude_blue magnitude_red -=============== =========== ======== =========== ============== ============= -0.00017862 8.3716516 6.857487 12.03939 12.56381 11.36282 -0.00031119 58.557541 2.518501 12.37415 12.81931 11.75581 -0.00058982 5.3805172 6.660454 12.95553 13.58890 12.19339 -0.00106097 -55.64880 7.893578 13.52746 14.63624 12.45566 -0.00113075 3.0206027 4.043843 12.74578 13.15043 12.17218 -... ... ... ... ... ... -=============== =========== ======== =========== ============== ============= - -Here is the meaning of the columns: - -0. ``right_ascension``: horizontal position of the star along the celestial equator (degrees) -#. ``declination``: vertical position of the star above/below celestial equator (degrees) -#. ``parallax``: amount the star's position in the sky changes throughout the year -#. ``magnitude_g``: how bright the star appears in our sky (inverse Logarithmic scale) -#. ``magnitude_blue``: blue-component of the color of the star -#. ``magnitude_red``: red-component of the color of the star +=========== ========= ============== +parallax magnitude color +=========== ========= ============== +0.006857487 12.03939 1.200986 +0.002518501 12.37415 1.063500 +0.006660454 12.95553 1.395508 +0.007893578 13.52746 2.180585 +0.004043843 12.74578 0.978254 +... ... ... +=========== ========= ============== Let's compute the distance to some well known stars in our night's sky. -For that, we need to extract the parallax measurements from the ``data`` array, which are in the column with index ``2``. +For that, we need to extract the parallax measurements from the ``data`` array, which are in the first column, the column with index ``0``. The way the array is laid out in memory makes it very fast to select portions of the data. We can index/slice arrays in a similar manner as Python lists using ``[index]``, but we do this separately along each dimension: ``[row_index, col_index]``:: - parallax = data[:, 2] # `:` is a shorthand for "select everything along this dimension" + parallax = data[:, 0] # `:` is a shorthand for "select everything along this dimension" print(parallax) Now for some basic math. -The parallax values are currently in `milli-arc-seconds `__. -We need to convert that into distance. -A common distance metric is the `"parsec" `__, which is the distance at which an object has a parallax of exactly one arc-second, which is 3.26156 light-years:: +We need to convert the parallax value into distance. +When we compute ``1 / parallax`` we get the distance in `"parsecs" `__ and each parsec is about 3.26156 light-years:: - distance_parsecs = 1 / (parallax / 1000) # 1000 milli-arc-seconds in an arc-second + distance_parsecs = 1 / parallax distance_ly = distance_parsecs * 3.26156 Note that we didn't write any ``for``-loops. @@ -154,94 +144,34 @@ Common mathematical operations include: ``+`` (:data:`numpy.add`), ``-`` (:data: Let's examine the closest star in the dataset:: - distance_to_closest_star = np.min(distance_ly) # returns the minimum value - index_of_closest_star = np.argmin(distance_ly) # returns the index of the minimum value - print(distance_to_closest_star, index_of_closest_star) + print(np.min(distance_ly)) The closest star is `Proxima Centauri `__ at 4.2465 light-years. -To see all the information we have on it, we can select the entire row using the index we just obtained:: +If we want to look up its magnitude, we would need to know which item in the ``distance_ly`` array was the smallest one:: - print(data[index_of_closest_star, :]) # select the row corresponding to Proxima Centauri + closest_star_index = np.argmin(distance_ly) + print(closest_star_index) + +That index we could use to select the proper row and column in the big ``data`` array:: + + print(data[index_of_closest_star, 1]) # column 1 contains the magnitude of the stars + +The magnitude scale works that the brightest stars have a magnitude of 1 and weaker stars have a magnitude of 2, 3 and so on. +So Proxima Centauri is quite bright in our night's sky! NumPy has many other ways to select data. For example, let's see if we can show some information on the 10 stars closest to us. To do this, we must first order the stars by distance:: - order_by_distance = np.argsort(distance_parsecs) # argsort returns indices, not values + order_by_distance = np.argsort(distance_ly) # argsort returns indices, not values closest_indices = order_by_distance[:10] # select the first 10 elements print(closest_indices) We now have a list (technically a 1D :class:`numpy.ndarray`) of integer indices of the stars closest to us. We can select multiple elements by indexing the array with such a list:: - print(data[closest_indices, :]) # selecting multiple rows in a 2D array - print(distance_ly[closest_indices]) # selecting multiple elements in a 1D array - -Can we find the `Big Dipper `__? - -.. figure:: img/numpy/big_dipper.svg - :width: 800px - :align: center - -**Left:** The Big Dipper as seen from Fujian. Image `retrieved from WikiMedia `__. **Right:** The Big Dipper on a sky chart. Image `retrieved from in-the-sky.org `__. Both images have a CC-BY license. - - -It lies within a section of the sky with a `right ascension `__ from 160 to 210 degrees, and a `declination `__ from 45 to 65 degrees:: - - right_ascension = data[:, 0] - declination = data[:, 1] - -To select elements from an array based on some condition, we can create an index based on boolean values. -Recall that in Python, comparison operations (``==``, ``<``, ``<=``, ``>``, ``>=``) produce a boolean (``True``/``False``) value. -See what happens when you apply them to an array:: - - print(right_ascension > 160) - -It produces a new array, of the same dimensions as the original array, but filled with ``True``/``False`` values indicating for each element whether the comparison holds true or not. -We call this a "boolean mask". -We can combine multiple masks using Python's boolean "and" operator ``&``:: - - boolean_mask = ( - (right_ascension > 160) & - (right_ascension < 210) & - (declination > 45) & - (declination < 65) - ) - -Once you have a boolean mask, you can apply it to any dimension of the array as long as it has the same number of elements as the mask:: - - sky_section = data[boolean_mask, :] - print(sky_section.shape) - -In our dataset, there are more than 15k stars in our chosen section of sky. -The Big Dipper should consist of the 7 brightest ones. -The column with index ``3``, named ``magnitude_g``, denotes how bright the star looks as seen by the Gaia sattelite. -The `magnitude scale `__ works "in reverse", such that very bright stars have a magnitude of 1 and weaker stars have a larger magnitude. - -.. code:: - - # Select the 7 brightest stars in our section of sky. - order_by_magnitude = np.argsort(sky_section[:, 3]) - brightest_stars = order_by_magnitude[:7] - big_dipper = sky_section[brightest_stars, :] - - # Make a figure with the position of the stars in the sky. You can take this at face - # value for now. We will cover how to make figures in a later lesson. - import matplotlib.pyplot as plt - fig, ax = plt.subplots() - ax.scatter(big_dipper[:, 0], big_dipper[:, 1]) - ax.xaxis.set_inverted(True) # right ascension goes from right-to-left - ax.set_aspect("equal") - ax.set_xlabel("right ascension (degrees)") - ax.set_ylabel("declination (degrees)") - - -Clever and efficient use of these indexing and mathematical operations is the key to NumPy's speed: you should try to cleverly use these selectors (written in C) to extract data to be used with other NumPy functions written in C or Fortran. -This will give you the benefits of Python with most of the speed of C. - -.. seealso:: - - :ref:`Numpy basic indexing docs ` + print(data[closest_indices, 1]) # magnitudes of the closest 10 stars + print(data[closest_indices, 2]) # color of the closest 10 stars Exercises 1 @@ -257,15 +187,31 @@ Exercises 1 What is the distance from us to Alpha Centauri AB? #. What is the distance from us to the brightest star in the dataset (`Sirius `__)? Remember that a smaller magnitude means a brighter star. - #. How many stars are there in the dataset that are even brighter than the brightest star in the Big Dipper (`Alioth `__)? + #. The stars in the Big Dipper are those with indices: ``[512739, 513978, 546586, 559674, 582978, 602054, 616331]``. + What are the distances of those stars to us? + Are they actually as close together as they seem? .. solution:: Solution Numpy-1 #. Row index: ``alpha_centauri_ind = np.argsort(distance_ly)[1]``. Its distance is ``distance_ly[alpha_centauri_ind]`` = 4.395 light-years. #. Brightest star index: ``brightest_ind = np.argsort(data[:, 3])[0]``. Its distance is ``distance_ly[brightest_ind]`` = 8.601 light-years. - #. The brightest star in the Big Dipper has a magnitude of ``np.min(big_dipper[:, 3])`` = 1.77. - There are ``data[data[:, 3] < np.min(big_dipper[:, 3])].shape`` = 28 stars in the dataset that are even brighter. + #. Distances of the Big Dipper stars: ``distance_ly[512739, 513978, 546586, 559674, 582978, 602054, 616331]``. + They are not as far apart from each other as you may think. + They actually move together through our Galaxy in a formation known as the `Ursa Major Moving Group `__. + +Get Orion, explore the distances of the stars of that constallation. + +Exercises 2 +----------- + +.. challenge:: Exercises: Numpy-2 + + #. How many stars are there in the dataset that are even brighter than the brightest star in Orion? (`Alioth `__)? + +.. solution:: Solution Numpy-2 + #. The brightest star in the Orion has a magnitude of ``np.min(orion[:, 1])`` = 1.77. + There are ``data[data[:, 1] < np.min(orion[:, 1])].shape`` = 28 stars in the dataset that are even brighter. Part 2: Estimating the distance of really far away stars -------------------------------------------------------- From 4ceb2fa63695c2e9e0f0965fd071d10c137412ca Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Thu, 15 Oct 2026 10:38:52 +0300 Subject: [PATCH 25/31] add big dipper image --- content/img/numpy/big_dipper.svg | 8882 ------------------------------ content/numpy-stars.rst | 4 + 2 files changed, 4 insertions(+), 8882 deletions(-) delete mode 100644 content/img/numpy/big_dipper.svg diff --git a/content/img/numpy/big_dipper.svg b/content/img/numpy/big_dipper.svg deleted file mode 100644 index 775baf14..00000000 --- a/content/img/numpy/big_dipper.svg +++ /dev/null @@ -1,8882 +0,0 @@ - -Ursa MajorUrsa MajorUrsa MajorUrsa MajorUrsa MajorUrsa MajorUrsa MajorUrsa MajorUrsa MajorUrsa MajorAliothAlkaidDubheMizarPhecdaMerakMegrezιhttps://in-the-sky.org45°50°55°60°45°50°55°210°150°165°180°195°60°right ascensiondeclination diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index 825497a4..6f478877 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -191,6 +191,10 @@ Exercises 1 What are the distances of those stars to us? Are they actually as close together as they seem? + .. figure:: img/numpy/Big_Dipper.jpg + :width: 300px + :align: center + .. solution:: Solution Numpy-1 #. Row index: ``alpha_centauri_ind = np.argsort(distance_ly)[1]``. Its distance is ``distance_ly[alpha_centauri_ind]`` = 4.395 light-years. 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a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -41,8 +41,8 @@ Let's explore the fundamentals of this foundational library by analyzing some as .. highlight:: python -Part 1: Exploring our local stellar neighbourhood -------------------------------------------------- +Part 1: Measuring distance through the parallax effect +------------------------------------------------------ The `Gaia sattelite `__ has imaged around 1.8 billion stars and other objects. Among the things it measured was the "parallax" of each object, which is how much it seems to "move" in relationship to very distant stars as the sattelite traveled around the sun and observed it from different angles. @@ -195,27 +195,16 @@ Exercises 1 :width: 300px :align: center + The Big Dipper as seen from Fujian. Image `retrieved from WikiMedia `__. CC-BY license. + .. solution:: Solution Numpy-1 #. Row index: ``alpha_centauri_ind = np.argsort(distance_ly)[1]``. Its distance is ``distance_ly[alpha_centauri_ind]`` = 4.395 light-years. #. Brightest star index: ``brightest_ind = np.argsort(data[:, 3])[0]``. Its distance is ``distance_ly[brightest_ind]`` = 8.601 light-years. - #. Distances of the Big Dipper stars: ``distance_ly[512739, 513978, 546586, 559674, 582978, 602054, 616331]``. - They are not as far apart from each other as you may think. - They actually move together through our Galaxy in a formation known as the `Ursa Major Moving Group `__. - -Get Orion, explore the distances of the stars of that constallation. - -Exercises 2 ------------ - -.. challenge:: Exercises: Numpy-2 + #. Distances of the Big Dipper stars: ``distance_ly[[512739, 513978, 546586, 559674, 582978, 602054, 616331]]`` = 84.49, 122.89, 83.18, 80.88, 82.55, 80.6, and 103.94 light years. + They are not as far apart from each other as you may think. + They actually move together through our Galaxy in a formation known as the `Ursa Major Moving Group `__. - #. How many stars are there in the dataset that are even brighter than the brightest star in Orion? (`Alioth `__)? - -.. solution:: Solution Numpy-2 - - #. The brightest star in the Orion has a magnitude of ``np.min(orion[:, 1])`` = 1.77. - There are ``data[data[:, 1] < np.min(orion[:, 1])].shape`` = 28 stars in the dataset that are even brighter. Part 2: Estimating the distance of really far away stars -------------------------------------------------------- @@ -229,9 +218,15 @@ When you are on the southern hemisphere of the Earth, you can see a massive clus The Large Magallanic Cloud. Photographed and edited by Martin Bernard, `retrieved from WikiMedia `__. CC BY-SA license. Its parallax is so small that even the Gaia satellite cannot reliably measure it, which means it must be pretty far away. -Still, we can estimate its distance, based on a striking relationship between the brightness and the color of a star. -A `Hertzsprung-Russel diagram `__ illustrates this. -Let's make one! +Still, we can estimate its distance, based on a striking relationship between the magnitude and the color of a star. +A `Hertzsprung-Russel diagram `__ illustrates this: + +.. figure:: img/numpy/hertzsprung_russel.jpg + :width: 400px + :align: center + + All of the stars in our dataset, plotted here as color versus intrinsic magnitude. + The main sequence is annotated as a green line. Recall that in our ``data`` array, the column with index ``3`` contains the measurements of the apparent brightness of each star according to Gaia's sensors (``magnitude_g``). For the Hertzsprung-Russel diagram, we need to estimate their true magnitude (luminosity), compensating for the fact that the further a star is, the dimmer it appears to us. @@ -347,7 +342,7 @@ The curve is not a straight line, but something like a fifth order polynomial sh .. note:: If you wish to skip the demonstration of how to perform linear algebra, you can use NumPy's build-in function to fit the polynomial function:: - beta = np.polyfit(bin_colors, 5) + beta = np.polyfit(bin_colors, bin_magnitudes, deg=5) A typical way to do this is a machine learning technique called `ordinary least squares (OLS) `__, where we collect everything we know in a matrix ``X`` and everything we wish to predict in a matrix ``Y`` and compute: From 9e29cc3c5e5a7f7bc5e7e34c43ecd1b0f7aea910 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Thu, 15 Oct 2026 13:40:24 +0300 Subject: [PATCH 27/31] working on lesson --- content/numpy-stars.rst | 139 +++++++++------------------------------- 1 file changed, 30 insertions(+), 109 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index 0a58b724..9b3c2dc8 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -70,12 +70,12 @@ We can use one of NumPy's functions to load the entire file into memory:: import numpy as np # we usually use `np` as shorthand # We set `skip_header=1` to skip the first line containing the column names. - data = np.genfromtxt("../resources/data/numpy/local_stars.csv", delimiter=",", skip_header=1) - print(data) + stars = np.genfromtxt("../resources/data/numpy/local_stars.csv", delimiter=",", skip_header=1) + print(stars) -The ``data`` variable is a NumPy array:: +The ``stars`` variable is a NumPy array:: - print(type(data)) + print(type(stars)) What is an array? ----------------- @@ -84,11 +84,11 @@ An array is a collection of values which are all of the same datatype, tightly p This is why we had to skip the first line of the file, the one containing ``str`` type descriptions, so that we were only reading numbers. In our case, the datatype of the resulting array is:: - print(data.dtype) + print(stars.dtype) Arrays can have any number of dimensions, ours has:: - print(data.ndim) + print(stars.ndim) You can think of a 1D array as a *list* and a 2D array as a *table*. Alternatively, in mathematics, we call a 1D array a *vector*, a 2D array a *matrix*, and an array with 3 or even more dimensions a *tensor*. @@ -97,7 +97,7 @@ Regardless of how many dimensions the array has, it is always a Python object of Let's see how many stars we have. The ``.shape`` property of an array contains the size along each dimension:: - print(data.shape) + print(stars.shape) We have information on more than a million stars, and for each star we have 6 values. @@ -119,11 +119,11 @@ parallax magnitude color =========== ========= ============== Let's compute the distance to some well known stars in our night's sky. -For that, we need to extract the parallax measurements from the ``data`` array, which are in the first column, the column with index ``0``. +For that, we need to extract the parallax measurements from the ``stars`` array, which are in the first column, the column with index ``0``. The way the array is laid out in memory makes it very fast to select portions of the data. We can index/slice arrays in a similar manner as Python lists using ``[index]``, but we do this separately along each dimension: ``[row_index, col_index]``:: - parallax = data[:, 0] # `:` is a shorthand for "select everything along this dimension" + parallax = stars[:, 0] # `:` is a shorthand for "select everything along this dimension" print(parallax) Now for some basic math. @@ -152,9 +152,9 @@ If we want to look up its magnitude, we would need to know which item in the ``d closest_star_index = np.argmin(distance_ly) print(closest_star_index) -That index we could use to select the proper row and column in the big ``data`` array:: +That index we could use to select the proper row and column in the big ``stars`` array:: - print(data[index_of_closest_star, 1]) # column 1 contains the magnitude of the stars + print(stars[index_of_closest_star, 1]) # column 1 contains the magnitude of the stars The magnitude scale works that the brightest stars have a magnitude of 1 and weaker stars have a magnitude of 2, 3 and so on. So Proxima Centauri is quite bright in our night's sky! @@ -170,8 +170,8 @@ To do this, we must first order the stars by distance:: We now have a list (technically a 1D :class:`numpy.ndarray`) of integer indices of the stars closest to us. We can select multiple elements by indexing the array with such a list:: - print(data[closest_indices, 1]) # magnitudes of the closest 10 stars - print(data[closest_indices, 2]) # color of the closest 10 stars + print(stars[closest_indices, 1]) # magnitudes of the closest 10 stars + print(stars[closest_indices, 2]) # color of the closest 10 stars Exercises 1 @@ -179,14 +179,7 @@ Exercises 1 .. challenge:: Exercises: Numpy-1 - Proxima Centauri is part of a 3-star system called `Alpha Centauri `__. - The other two stars are very close together and appear as a single object (Alpha Centauri AB) in this dataset. - #. What is the row index of the binary star Alpha Centauri AB? - It is the second closest star to us. - What is the distance from us to Alpha Centauri AB? - #. What is the distance from us to the brightest star in the dataset (`Sirius `__)? - Remember that a smaller magnitude means a brighter star. #. The stars in the Big Dipper are those with indices: ``[512739, 513978, 546586, 559674, 582978, 602054, 616331]``. What are the distances of those stars to us? Are they actually as close together as they seem? @@ -195,15 +188,23 @@ Exercises 1 :width: 300px :align: center - The Big Dipper as seen from Fujian. Image `retrieved from WikiMedia `__. CC-BY license. + The Big Dipper as seen from Fujian. Image `retrieved from WikiMedia `__. CC-BY license. + #. Proxima Centauri is part of a 3-star system called `Alpha Centauri `__. + The other two stars are very close together and appear as a single object (Alpha Centauri AB) in this dataset. + + #. What is the row index of the binary star Alpha Centauri AB? + It is the second closest star to us. + #. What is the distance from us to Alpha Centauri AB? + #. What is the distance from us to the brightest star in the dataset (`Sirius `__)? + Remember that a smaller magnitude means a brighter star. .. solution:: Solution Numpy-1 - #. Row index: ``alpha_centauri_ind = np.argsort(distance_ly)[1]``. Its distance is ``distance_ly[alpha_centauri_ind]`` = 4.395 light-years. - #. Brightest star index: ``brightest_ind = np.argsort(data[:, 3])[0]``. Its distance is ``distance_ly[brightest_ind]`` = 8.601 light-years. #. Distances of the Big Dipper stars: ``distance_ly[[512739, 513978, 546586, 559674, 582978, 602054, 616331]]`` = 84.49, 122.89, 83.18, 80.88, 82.55, 80.6, and 103.94 light years. They are not as far apart from each other as you may think. They actually move together through our Galaxy in a formation known as the `Ursa Major Moving Group `__. + #. Row index: ``alpha_centauri_ind = np.argsort(distance_ly)[1]``. Its distance is ``distance_ly[alpha_centauri_ind]`` = 4.395 light-years. + #. Brightest star index: ``brightest_ind = np.argsort(stars[:, 3])[0]``. Its distance is ``distance_ly[brightest_ind]`` = 8.601 light-years. Part 2: Estimating the distance of really far away stars @@ -222,104 +223,25 @@ Still, we can estimate its distance, based on a striking relationship between th A `Hertzsprung-Russel diagram `__ illustrates this: .. figure:: img/numpy/hertzsprung_russel.jpg - :width: 400px + :width: 500px :align: center All of the stars in our dataset, plotted here as color versus intrinsic magnitude. The main sequence is annotated as a green line. + Other than the main sequence stars, you can see the red giants branching off at the top, bright but yellow/red, and the white dwarfs at the bottom, very dim but blue/white. -Recall that in our ``data`` array, the column with index ``3`` contains the measurements of the apparent brightness of each star according to Gaia's sensors (``magnitude_g``). -For the Hertzsprung-Russel diagram, we need to estimate their true magnitude (luminosity), compensating for the fact that the further a star is, the dimmer it appears to us. -We will use the formula given in an `official Gaia paper `__ for this:: +Based on this chart, we can derive a formula to translate the color of a main sequence star (so not a red giant or white dwarf) to its intrinsic magnitude, that is: how bright the star would appear to us if it were only a single parsec away from us. - magnitude_g = data[:, 3] # apparent magnitude - magnitude = magnitude_g + 5 * np.log10(parallax) - 10 # true magnitude - -The :data:`numpy.log10` function is part of NumPy's vast collection of `ufuncs `__, which are functions that operate on each element of an array. - -In our diagram, we want to show the color of a star on a scale from "very blue" to "very red", with yellow stars in between. -We can compute this by taking the difference between the blue and red components of its color:: - - magnitude_blue = data[:, 4] - magnitude_red = data[:, 5] - color = magnitude_blue - magnitude_red - -Now we have everything we need to make the plot. Again, just take this plotting code at face value for now:: - - import matplotlib.pyplot as plt - from matplotlib.colors import LinearSegmentedColormap - plt.style.use("dark_background") - plt.figure(figsize=(6, 8)) - cmap = LinearSegmentedColormap.from_list( - "stars", ["#ccfeff", "#ffffff", "#fff18f", "#e5d000", "#dd6a00", "#cc0000", "#6b0000"] - ) - plt.scatter(color, magnitude, s=1, c=color, cmap=cmap, alpha=0.1, clim=(0, 3)) - plt.gca().invert_yaxis() # put bright stars on top, dim stars at the bottom - plt.title("Hertzsprung-Russell diagram") - plt.xlabel("Color (Blue ⇨ Red)") - plt.ylabel("Magnitude") - plt.tight_layout() +.. math:: -Other than the main sequence stars, you can see the red giants branching off at the top, bright but red, and the white dwarfs at the bottom, very dim but white. + \text{magnitude} = \beta_5\,\text{color}^5 + \beta_4\,\text{color}^4 + \beta_2\,\text{color}^3 + \beta_2\,\text{color}^2 + \beta_1\,\text{color} + \beta_0 -Based on this chart, we can derive a formula to translate the color of a main sequence star (so not a red giant or white dwarf) to its true brightness. -Then, we can compare that true brightness with its apparent brightness to deduce how much dimmer it appears to us than it actually is, hence how far away it must be. +Then, we can compare that intrinsic magnitude with its apparent magnitude to deduce how much dimmer it appears to us than it actually is, hence how far away it must be. Creating arrays --------------- -The first step is to extract a bunch of ``(color, magnitude)`` pairs along the main sequence, free from outliers. -The strategy is to divide the color spectrum into 100 bins and take the median magnitude in each bin. -With the vast majority of the stars being main sequence stars, the median should be very representative of the general curve of the main sequence. - -There are various ways to create an array with equally spaced numbers. -For example, there is the NumPy equivalent of Python's :func:`range` function, :func:`numpy.arange`:: - - bins = np.arange(0.1, 4.5, step=0.04) - -But for our purposes, :func:`numpy.linspace` is even better:: - - n_bins = 100 - bins = np.linspace(0.1, 4.5, num=n_bins) # 100 numbers evenly spread between 0.1 and 4.5 - -We could assign stars to their respective color bins using boolean masking, but NumPy offers the :func:`numpy.digitize` function especially for this purpose:: - - bin_assignment = np.digitize(color, bins) - print(bin_assignment) - -The function gives for each element of the ``color`` array, the index of the bin it belongs to. -Since each element of the ``color`` array is the color of a star, we have effectively assigned each star to a bin. -Now, we can compute for each bin, the median brightness and color of all the stars in that bin. -We will do this using a ``for`` loop and use a typical pattern for creating NumPy arrays: we first collect the values into a Python list and then convert that list into a NumPy array:: - - # We collect the median colors and magnitudes in these lists. - bin_colors = list() - bin_magnitudes = list() - - # Loop over all the bins and compute the median color/magnitude. - for bin_index in range(n_bins): - bin_color = np.median(color[bin_assignment == bin_index]) - bin_magnitude = np.median(magnitude[bin_assignment == bin_index]) - bin_colors.append(bin_color) - bin_magnitudes.append(bin_magnitude) - - # Convert the Python lists to NumPy arrays. - bin_colors = np.array(bin_colors) - bin_magnitudes = np.array(bin_magnitudes) - - # The white dwarf stars with a color < 0.1 are all lumped in the first bin, - # so that bin is unreliable, chop it off. - bin_colors = bin_colors[1:] - bin_magnitudes = bin_magnitudes[1:] - -.. note:: - Appending values to a Python list is very fast, but appending values to a NumPy array is very slow. This is why we first collect the values in a Python list and then convert it to a NumPy array. - -.. seealso:: - - `Numpy array creation docs `_ - Doing linear algebra -------------------- @@ -342,7 +264,6 @@ The curve is not a straight line, but something like a fifth order polynomial sh .. note:: If you wish to skip the demonstration of how to perform linear algebra, you can use NumPy's build-in function to fit the polynomial function:: - beta = np.polyfit(bin_colors, bin_magnitudes, deg=5) A typical way to do this is a machine learning technique called `ordinary least squares (OLS) `__, where we collect everything we know in a matrix ``X`` and everything we wish to predict in a matrix ``Y`` and compute: From b92133476a7d6ae88c63d6a9d127dc3f73c2f6f8 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Fri, 16 Oct 2026 08:55:20 +0300 Subject: [PATCH 28/31] add coeffs --- content/numpy-stars.rst | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index 9b3c2dc8..7877c574 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -234,7 +234,8 @@ Based on this chart, we can derive a formula to translate the color of a main se .. math:: - \text{magnitude} = \beta_5\,\text{color}^5 + \beta_4\,\text{color}^4 + \beta_2\,\text{color}^3 + \beta_2\,\text{color}^2 + \beta_1\,\text{color} + \beta_0 + \text{magnitude} = 0.9519014\,\text{color}^5 + 4.24472718\,\text{color}^4 + 0.65014753\,\text{color}^3 + -0.87512072\,\text{color}^2 + 0.27781372\,\text{color} + -0.02884251 + Then, we can compare that intrinsic magnitude with its apparent magnitude to deduce how much dimmer it appears to us than it actually is, hence how far away it must be. From 765e755257835a39c05f5f3e16e7f1ecaed54f84 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Fri, 16 Oct 2026 10:59:05 +0300 Subject: [PATCH 29/31] finish first round of simplifications --- content/numpy-stars.rst | 211 +++++++++++++--------------------------- 1 file changed, 70 insertions(+), 141 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index 9b3c2dc8..78c3fd4f 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -54,7 +54,7 @@ We can use the amount of parallax to determine how far away a star is. Visual explanation of the parallax effect. Image `retrieved from WikiMedia `__. Public domain. -A sample of the Gaia data can be found in :download:`../resources/data/numpy/local_stars.csv`. +A sample of the Gaia data can be found in ``local_stars.csv``. This is a *comma separated value (CSV)* file, where each line contains the parallax, magnitude and color of the star. The first line of that file is the "header" containing the column names:: @@ -69,8 +69,12 @@ The first line of that file is the "header" containing the column names:: We can use one of NumPy's functions to load the entire file into memory:: import numpy as np # we usually use `np` as shorthand + + # Set this to the location where you unzipped the data package. + stars_filename = "../resources/data/numpy/local_stars.csv" + # We set `skip_header=1` to skip the first line containing the column names. - stars = np.genfromtxt("../resources/data/numpy/local_stars.csv", delimiter=",", skip_header=1) + stars = np.genfromtxt(stars_filename, delimiter=",", skip_header=1) print(stars) The ``stars`` variable is a NumPy array:: @@ -203,8 +207,10 @@ Exercises 1 #. Distances of the Big Dipper stars: ``distance_ly[[512739, 513978, 546586, 559674, 582978, 602054, 616331]]`` = 84.49, 122.89, 83.18, 80.88, 82.55, 80.6, and 103.94 light years. They are not as far apart from each other as you may think. They actually move together through our Galaxy in a formation known as the `Ursa Major Moving Group `__. - #. Row index: ``alpha_centauri_ind = np.argsort(distance_ly)[1]``. Its distance is ``distance_ly[alpha_centauri_ind]`` = 4.395 light-years. - #. Brightest star index: ``brightest_ind = np.argsort(stars[:, 3])[0]``. Its distance is ``distance_ly[brightest_ind]`` = 8.601 light-years. + #. Row index: ``alpha_centauri_ind = np.argsort(distance_ly)[1]`` = 650564. + Its distance is ``distance_ly[alpha_centauri_ind]`` = 4.395 light-years. + #. Brightest star index: ``brightest_ind = np.argsort(stars[:, 1])[0]`` = 303978. + Its distance is ``distance_ly[brightest_ind]`` = 8.601 light-years. Part 2: Estimating the distance of really far away stars @@ -218,119 +224,74 @@ When you are on the southern hemisphere of the Earth, you can see a massive clus The Large Magallanic Cloud. Photographed and edited by Martin Bernard, `retrieved from WikiMedia `__. CC BY-SA license. -Its parallax is so small that even the Gaia satellite cannot reliably measure it, which means it must be pretty far away. -Still, we can estimate its distance, based on a striking relationship between the magnitude and the color of a star. + +The Gaia data for a collection of stars that are part of the Large Magallanic Cloud can be found in ``lmc_stars.rst``:: + + # Set this to where you unzipped the data package + lmc_filename = "../resources/data/numpy/lmc_stars.csv" + lmc_stars = np.genfromtxt(lmc_filename, delimiter=",", skip_header=1) + print(lmc_stars) + +Just as before, the first column of the file contains the parallax, but in this case they are all 0. +The parallax of these stars is so small that even the Gaia satellite cannot reliably measure it, which means they must be pretty far away. +Still, we can estimate their distances, based on a striking relationship between the magnitude and the color of a star. A `Hertzsprung-Russel diagram `__ illustrates this: .. figure:: img/numpy/hertzsprung_russel.jpg :width: 500px :align: center - All of the stars in our dataset, plotted here as color versus intrinsic magnitude. + All of the stars in our main dataset, plotted here as color versus intrinsic magnitude. The main sequence is annotated as a green line. Other than the main sequence stars, you can see the red giants branching off at the top, bright but yellow/red, and the white dwarfs at the bottom, very dim but blue/white. Based on this chart, we can derive a formula to translate the color of a main sequence star (so not a red giant or white dwarf) to its intrinsic magnitude, that is: how bright the star would appear to us if it were only a single parsec away from us. +The curve is not a straight line, but this fifth order polynomial is a decent fit: .. math:: - \text{magnitude} = \beta_5\,\text{color}^5 + \beta_4\,\text{color}^4 + \beta_2\,\text{color}^3 + \beta_2\,\text{color}^2 + \beta_1\,\text{color} + \beta_0 - -Then, we can compare that intrinsic magnitude with its apparent magnitude to deduce how much dimmer it appears to us than it actually is, hence how far away it must be. - - -Creating arrays ---------------- - - -Doing linear algebra --------------------- - -If we fit a curve through the ``(color, magnitude)`` combinations, we obtain a function to predict a star's magnitude based on its color. -We can do this with a bit of linear algebra. -To get a feel for how the curve should look, let's draw the ``(color, magnitude)`` combinations on top of the Hertzsprung-Russel diagram (assuming it is still open, re-create it if it isn't):: - - plt.plot(bin_colors, bin_magnitudes, color="green", linewidth=2) - -The curve is not a straight line, but something like a fifth order polynomial should be pretty good fit: - -.. math:: - :name: A fifth order polynomial function - - \text{magnitude} = \beta_5\,\text{color}^5 + \beta_4\,\text{color}^4 + \beta_2\,\text{color}^3 + \beta_2\,\text{color}^2 + \beta_1\,\text{color} + \beta_0 - -"Fitting the curve" now means choosing the optimal :math:`\beta` values so that when we input the ``color``, we get a good prediction of ``magnitude``. - -.. note:: - If you wish to skip the demonstration of how to perform linear algebra, you can use NumPy's build-in function to fit the polynomial function:: - + \text{magnitude} = -0.29\,\text{color}^5 + 0.28\,\text{color}^4 - 0.88\,\text{color}^3 + 0.65\,\text{color}^2 + 4.2\,\text{color} + 0.95 -A typical way to do this is a machine learning technique called `ordinary least squares (OLS) `__, where we collect everything we know in a matrix ``X`` and everything we wish to predict in a matrix ``Y`` and compute: +To show off NumPy's linear algebra capabilities, we will compute this equation with the following vector-matrix product: .. math:: - :name: The ordinary least squares function - - \beta = (X^T X)^{-1} X^T Y - -The resulting ``beta`` vector contains the optimal linear combination (in the least-squares sense) of the columns of ``X`` to predict the columns of ``Y``. -So, if we create a 2D array ``X`` where the columns are ``bin_colors`` raised to different powers, and set ``Y`` to ``bin_magnitude``, we should obtain the beta values we need. -The code below does this, while demonstrating how to create a 2D array by gluing 1D arrays together, and how to generate an array containing only ``1``'s:: - - X = np.column_stack(( - np.ones_like(bin_colors), # bin_colors ** 0 - bin_colors, # bin_colors ** 1 - bin_colors ** 2, - bin_colors ** 3, - bin_colors ** 4, - bin_colors ** 5, - )) - Y = bin_magnitudes - -To compute the OLS formula, we need a bunch of linear algebra functionality: -* matrix multiplication, which is written as ``@`` in Python -* matrix transpose, which NumPy arrays support through their property ``.T`` -* matrix inversion, which is done through :func:`numpy.linalg.inv` + \text{magnitude} = + \begin{bmatrix} + -0.028, 0.28, -0.88, 0.65, 4.2, 0.95 + \end{bmatrix} + \cdot + \begin{bmatrix} + \text{color}^5 \\ \text{color}^4 \\ \text{color}^3 \\ \text{color}^2 \\ \text{color} \\ 1 + \end{bmatrix} -Using NumPy, the Python code can look very much like the math formula:: - beta = np.linalg.inv(X.T @ X) @ X.T @ Y - print(beta) - -.. seealso:: - `Numpy linear algebra functionality `__. +Creating arrays +--------------- +We use the :func:`numpy.array` function to create NumPy arrays. +First the ``coefficients`` vector:: -Creating and documenting functions in the scientific computing style --------------------------------------------------------------------- + coefficients = np.array([-0.028, 0.28, -0.88, 0.65, 4.2, 0.95]) + print(coefficients.shape) -Now that we have ``beta``, we can use formula (1) to predict a star's magnitude given its color. -Let's wrap the formula in a function, so we can give it a name and write documentation on how to use it. -NumPy comes with a style guide for writing docstrings for functions called `NumPyDoc `__:: +And now the matrix containing on each row the colors of the stars, raised to some power:: - def predict_magnitude(color, beta): - """Predict a main sequence star's inherent magnitude given its color. + color = lmc_stars[:, 2] # this column contains the color of the stars + color_powers = np.array([ + color ** 5, + color ** 4, + color ** 3, + color ** 2, + color, + np.ones_like(color), # produces an array filled with 1's + ]) + print(color_powers.shape) - Parameters - ---------- - color : float | array of float, shape (n_stars,) - The color of the star(s), computed as magnitude_blue - magnitude_red. - beta: array of float, shape (6,) - The coefficients describing the main-sequence curve. +We estimate the intrinsic magnitude of our Large Magallanic Cloud stars by using a matrix multiplication between the ``coefficients`` vector and the ``color_powers`` matrix:: - Returns - ------- - magnitude : float - The estimated true magnitude of the star(s). - """ - return ( - beta[5] * color ** 5 + - beta[4] * color ** 4 + - beta[3] * color ** 3 + - beta[2] * color ** 2 + - beta[1] * color + - beta[0] - ) + intrinsic_magnitude = coefficients @ color_powers # `@` means matrix multiplication + print(intrinsic_magnitude) Exercises 2 @@ -338,67 +299,35 @@ Exercises 2 .. challenge:: Exercises: Numpy-2 - In this exercise, you will estimate the distance to the Large Magallanic Cloud. - - We have shown you how to estimate the true magnitude of a star based on its color and we wrote the ``predict_magnitude`` function to do it. - To predict distance, you can compare a star's estimated true magnitude with its apparent magnitude on Gaia's sensors. - The further away the star is, the dimmer it appears to Gaia. - Recall the formula we used earlier:: - - magnitude = magnitude_g + 5 * np.log10(parallax) - 10 - - When we solve for ``parallax``, we get:: - - estimated_parallax = np.pow(10, (magnitude - magnitude_g + 10) / 5) - - 1. Write a function to predict the distance of a star given its apparent magnitude, its color, and the ``beta`` coefficients we computed. The function signature should be:: + In this exercise, you will estimate the distance to the Large Magallanic Cloud (LMC). - predict_distance(color, magnitude_g, beta) + #. Compare the intrinsic magnitude of the LMC stars as we computed together with their apparent magnitude (how bright they appear on Gaia's sensors: ``lmc_stars[:, 1]``) to deduce how much dimmer they appear to us than they actually are, hence how far away they must be. + The formula for this is as follows:: - 1. Use the ``predict_magnitude`` function to estimate the true magnitude. - 2. Use the formula above to convert the difference between the apparent magnitude (``magnitude_g``) and true magnitude (``magnitude``) to a parallax value. 3. Convert the parallax value to a distance in light-years. See the lesson material above on how to do this if you don't remember. + parallax = 10 ** ((intrinsic_magnitude - apparent_magnitude - 5) / 5) - 2. Some main sequence stars from the Large Magallanic Cloud can be found in :download:`../resources/data/numpy/lmc_stars.csv`. - To read it, use :func:`numpy.genfromtext` in the same manner as we did with the ``data`` array. - Predict their distance using the function you just created and take the average (:func:`numpy.mean`) as a representative value for roughly how far away the cloud is. + #. Convert the ``parallax`` you obtained with the above formula to distance in lightyears. + See Part 1 of this lesson on how to do this. + #. Use the average distance (:func:`numpy.mean`) as a representative value for roughly how far away the cloud is. Given that the `Milky Way `__ galaxy is about 87400 light-years across, what does the distance to the Large Magallanic Cloud tell us? .. solution:: Solution Numpy-2 - #. .. code:: + #. .. code-block:: - def predict_distance(color, magnitude_g, beta): - """Predict a main sequence star's distance given its apparent magnitude and color. + apparent_magnitude = lmc_stars[:, 1] + lmc_parallax = 10 ** ((intrinsic_magnitude - apparent_magnitude - 5) / 5) - Parameters - ---------- - magnitude_g : float | array of float, shape (n_stars,) - The apparent magnitude of the star(s). - color : float | array of float, shape (n_stars,) - The color of the star(s), computed as magnitude_blue - magnitude_red. - beta: array of float, shape (6,) - The coefficients describing the main-sequence curve. + #. .. code-block:: - Returns - ------- - distance : float - The estimated distance in light-years. - """ - magnitude = predict_magnitude(color, beta) - parallax = np.pow(10, (magnitude - magnitude_g + 10) / 5) - distance_parsecs = 1 / (parallax / 1000) - return distance_parsecs * 3.26156 # convert to light-years + lmc_distance_ly = (1 / lmc_parallax) * 3.26156 - #. .. code :: + #. .. code-block:: - lmc = np.genfromtxt( - "../resources/data/numpy/lmc_stars.csv", delimiter=",", skip_header=1 - ) - distance = predict_distance(lmc[:, 3], lmc[:, 4] - lmc[:, 5], beta) - print(np.mean(distance.mean)) + print(np.mean(lmc_distance_ly)) - The Large Magallanic Cloud is roughly 158000 light-years away. - This means it does not lie in our galaxy, but must be a galaxy of its own. + The Large Magallanic Cloud is roughly 159000 light-years away. + This means it does not lie in our galaxy, but must be a galaxy of its own. Where to go from here @@ -413,5 +342,5 @@ The Gaia satellite did not image the brightest stars, the most famous ones, beca Hence, the 50 brightest stars our dataset were not from Gaia, but were instead taken from the SIMBAD database. Those are the stars we look at during the Big Dipper demo. -For the Large Magallanic Cloud exercise, we cherry picked the stars in :download:`../resources/data/numpy/lmc_stars.csv` to be perfectly on the main sequence and thus ensuring that you get a decent estimate of the distance. +For the Large Magallanic Cloud exercise, we cherry picked the stars in ``lmc_stars.csv`` to be perfectly on the main sequence and thus ensuring that you get a decent estimate of the distance. In reality it's quite an ordeal to choose the proper stars to base your distance estimate on. From 80a20e88868a8fed476b7ac7089253835951af85 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Fri, 16 Oct 2026 11:21:35 +0300 Subject: [PATCH 30/31] tweaks --- content/numpy-stars.rst | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index 78c3fd4f..3dbf7481 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -6,9 +6,9 @@ NumPy .. questions:: - Why and when should I use NumPy? + - Why is NumPy faster than plain Python? - How do we frame scientific questions in terms of NumPy computations? - How do I translate a mathematical formula into a line of NumPy code? - - Why is NumPy faster than plain Python? .. objectives:: @@ -286,7 +286,7 @@ And now the matrix containing on each row the colors of the stars, raised to som color, np.ones_like(color), # produces an array filled with 1's ]) - print(color_powers.shape) + print(color_powers) We estimate the intrinsic magnitude of our Large Magallanic 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:class: demo Each lesson is placed inside an example scientific setting. In this lesson, we will compute the distance of the stars in our night sky, using the data collected by the Gaia sattelite. + To follow along, you must download the :download:`data package ` and unzip it in the ``python-for-scicomp`` folder. NumPy is the most used library for scientific computing. Even if you are not using it directly, chances are high that some library uses it in the background. @@ -32,7 +30,7 @@ It helps you work with "data", as in large amounts of numbers, by providing: 2. Efficient routines to manipulate data, such as reading and writing, splitting and concatenation, and other transformations 3. 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z6}+tOFUruNC9P@g<1>>h=Wv<}Ta5)9j2?a57hSv7pDM#$k&b+Y1)?O4l9BY7;OnuP z_{yh>VT*^&GKe_iflUCXhX0k4W5+@}d?s@ulsRc@nqz$xTi)ck$J6T*hj)n=OdntM zW1$y`%7+E&PjB(O?93BxwtMMWFBlH>+>zb!dbv!fq#V}iuM8eO7k`W4O)6RnY4ulN zqG;=@NtF>>=r*wuDJLFf8FUo(qoq%eY8Uobebs9oK-t0T^HQV3yy``5N9`t;VRV z9AhHAe%T!V&4PfgYasQ1&Hmk)#e-%K(4lS$|BoB;^qq~%|7P?I1f3)QC-iqVJ*>L9 zT?dDEYR>BqN>%|kbA1_!UwE!RKQj-s=R->YJY%C`{5|kr=-C%qsHUCzU6w~%~sw}~b@01@Clyn6ZK-X-ec281fuMj|pXU@|FwX4a0_jzmcy zn<73=MbiHcL`7*dxXoY=*z*VGA7S&C!P0DGNwCFglM5wC<{*aynQOr_5%9=TR{xLJ z*9pjLZ&AcZP-q6?bGAqt>jIv|(tCCwy#I6FG|JEP@7}WhXP(da+SY|x1>hC9(=$(A zu;amd%%O5KB@GcGQuU~McXVR7S+W;Lc0dRF6UK}W`uA~<5C`w!VK3GWxE@LlWx>rq ReftGYnHt?Nth(kL`7eIYUhn__ From 37f733db525d09651e7e84b257dee9d785db3df1 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Sun, 5 Jul 2026 11:45:17 +0300 Subject: [PATCH 13/31] Disable PDF build until it can be fixed --- .github/workflows/sphinx.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/.github/workflows/sphinx.yml b/.github/workflows/sphinx.yml index a0aa4a14..39198094 100644 --- a/.github/workflows/sphinx.yml +++ b/.github/workflows/sphinx.yml @@ -15,7 +15,7 @@ env: # If these SPHINXOPTS are enabled, then be strict about the # builds and fail on any warnings. #SPHINXOPTS: "-W --keep-going -T" - GENERATE_PDF: true # to enable, must be 'true' lowercase + GENERATE_PDF: false # to enable, must be 'true' lowercase GENERATE_SINGLEHTML: true # to enable, must be 'true' lowercase PDF_FILENAME: lesson.pdf MULTIBRANCH: true # to enable, must be 'true' lowercase From 664d33c8000a7b9d5fdf40e4cb086453e65f63a1 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Sun, 5 Jul 2026 11:53:57 +0300 Subject: [PATCH 14/31] Make Github actions expose the HTML lession as an artifact --- .github/workflows/sphinx.yml | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/.github/workflows/sphinx.yml b/.github/workflows/sphinx.yml index 39198094..e69c80b8 100644 --- a/.github/workflows/sphinx.yml +++ b/.github/workflows/sphinx.yml @@ -96,6 +96,12 @@ jobs: make singlehtml mv _build/singlehtml/ _build/dirhtml/singlehtml/ + - name: Save HTML as artifact + uses: actions/upload-artifact@v4 + with: + name: html-dev + path: _build/dirhtml + # PDF if requested - name: Generate PDF if: ${{ env.GENERATE_PDF == 'true' }} From e06206c3fa3d27cf033147a8d706c08ec647a6a7 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Sun, 5 Jul 2026 11:56:11 +0300 Subject: [PATCH 15/31] Update HTML artifact --- .github/workflows/sphinx.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/.github/workflows/sphinx.yml b/.github/workflows/sphinx.yml index e69c80b8..f82f8a06 100644 --- a/.github/workflows/sphinx.yml +++ b/.github/workflows/sphinx.yml @@ -100,7 +100,7 @@ jobs: uses: actions/upload-artifact@v4 with: name: html-dev - path: _build/dirhtml + path: _build # PDF if requested - name: Generate PDF From e2df9188550a930832dfb1ed79f78f732831a0cb Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Sun, 5 Jul 2026 12:01:31 +0300 Subject: [PATCH 16/31] Change title --- content/numpy-stars.rst | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index 1bb02908..3450038d 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -1,7 +1,7 @@ .. _numpy_stars: -NumPy Stars -=========== +NumPy +===== .. questions:: From 8aa1eb730b7f31ef726c0ec75556859a32c29b77 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Sun, 11 Oct 2026 10:29:10 +0300 Subject: [PATCH 17/31] Some tweaks to the numpy stars lesson --- content/numpy-stars.rst | 17 ++++++++++++----- 1 file changed, 12 insertions(+), 5 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index 3450038d..6b734963 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -365,11 +365,17 @@ To get a feel for how the curve should look, let's draw the ``(color, magnitude) The curve is not a straight line, but something like a fifth order polynomial should be pretty good fit: .. math:: - :name: A fifth order polynonial function + :name: A fifth order polynomial function \text{magnitude} = \beta_5\,\text{color}^5 + \beta_4\,\text{color}^4 + \beta_2\,\text{color}^3 + \beta_2\,\text{color}^2 + \beta_1\,\text{color} + \beta_0 "Fitting the curve" now means choosing the optimal :math:`\beta` values so that when we input the ``color``, we get a good prediction of ``magnitude``. + +.. note:: + If you wish to skip the demonstration of how to perform linear algebra, you can use NumPy's build-in function to fit the polynomial function:: + + beta = np.polyfit(bin_colors, 5) + A typical way to do this is a machine learning technique called `ordinary least squares (OLS) `__, where we collect everything we know in a matrix ``X`` and everything we wish to predict in a matrix ``Y`` and compute: .. math:: @@ -456,11 +462,12 @@ Exercises 2 estimated_parallax = np.pow(10, (magnitude - magnitude_g + 10) / 5) - 1. Write a function to predict the distance of a star given its apparent magnitude, its color, and the ``beta`` coefficients we computed: + 1. Write a function to predict the distance of a star given its apparent magnitude, its color, and the ``beta`` coefficients we computed. The function signature should be:: + + predict_distance(color, magnitude_g, beta) 1. Use the ``predict_magnitude`` function to estimate the true magnitude. - 2. Use the formula above to convert the difference between the apparent magnitude (``magnitude_g``) and true magnitude (``magnitude``) to a parallax value. - 3. Convert the parallax value to a distance in light-years. See the lesson material above on how to do this if you don't remember. + 2. Use the formula above to convert the difference between the apparent magnitude (``magnitude_g``) and true magnitude (``magnitude``) to a parallax value. 3. Convert the parallax value to a distance in light-years. See the lesson material above on how to do this if you don't remember. 2. Some main sequence stars from the Large Magallanic Cloud can be found in :download:`../resources/data/numpy/lmc_stars.csv`. To read it, use :func:`numpy.genfromtext` in the same manner as we did with the ``data`` array. @@ -499,7 +506,7 @@ Exercises 2 "../resources/data/numpy/lmc_stars.csv", delimiter=",", skip_header=1 ) distance = predict_distance(lmc[:, 3], lmc[:, 4] - lmc[:, 5], beta) - print(distance.mean()) + print(np.mean(distance.mean)) The Large Magallanic Cloud is roughly 158000 light-years away. This means it does not lie in our galaxy, but must be a galaxy of its own. From 08ef039a073db9598bdd95969e148dce44fc1706 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Mon, 12 Oct 2026 12:57:22 +0300 Subject: [PATCH 18/31] Some improvements based in Richards comments --- content/numpy-stars.rst | 84 ++++++++++++++++++++++++++++++----------- 1 file changed, 61 insertions(+), 23 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index 6b734963..a870d33f 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -5,24 +5,30 @@ NumPy .. questions:: - - Why use NumPy instead of pure python? - - How to use basic NumPy? - - What is vectorization? + - Why and when should I use NumPy? + - How do we frame scientific questions in terms of NumPy computations? + - How do I translate a mathematical formula into a line of NumPy code? + - Why is NumPy faster than plain Python? .. objectives:: - - Understand the Numpy array object - - Be able to use basic NumPy functionality - - Understand enough of NumPy to search for answers to the rest of your questions ;) + This is not a complete NumPy tutorial. Instead, this lesson strives to teach you: + - How to use basic NumPy: arrays, selecting data, performing mathematical operations + - A sense of how NumPy code looks: vectorized code. + - To understand enough of NumPy to search for answers to the rest of your questions. ;) .. admonition:: Example context :class: demo - Each lesson is placed inside an example scientific setting. - In this lesson, we will compute the distance of the stars in our night sky, using the data collected by the Gaia sattelite. + This lesson is placed inside an example scientific setting. + We will compute the distance of the stars in our night sky, using the data collected by the Gaia sattelite. To follow along, you must download the :download:`data package ` and unzip it in the ``python-for-scicomp`` folder. +Why NumPy? +---------- + NumPy is the most used library for scientific computing. +If NumPy did not exist, Python would not be such a popular language for scientific computing, because it would be too slow. Even if you are not using it directly, chances are high that some library uses it in the background. It helps you work with "data", as in large amounts of numbers, by providing: @@ -51,23 +57,46 @@ A sample of the Gaia data can be found in :download:`../resources/data/numpy/loc This is a *comma separated value (CSV)* file, where each line contains 6 numbers describing a single object. The first line of that file is the "header" indicating the meaning of each number:: - with open("../resources/data/numpy/local_stars.csv") as f: - # The file handle `f` acts as a generator that yields lines of text. - first_line = next(f) # take a single item from the generator - names = first_line.strip().split(",") # get a list of individual names - print(names) + right_ascension,declination,parallax,magnitude_g,magnitude_blue,magnitude_red + 0.0001786275254186,8.371651684580781,6.857487064725919,12.039392,12.56381,11.3628235 + 0.0003111955790572,58.5575410957825,2.5185016849859228,12.374151,12.819315,11.755815 + 0.0005898233877197,5.380517264865798,6.660454655312786,12.95553,13.5889015,12.193393 + 0.0010609752955102,-55.64880544379682,7.89357866196891,13.52746,14.636247,12.455662 + 0.0011307510147073,3.0206027981262924,4.043843423571794,12.745788,13.150434,12.17218 + ... + +Here is the data, formatted as a table: + +=============== =========== ======== =========== ============== ============= +right_ascension declination parallax magnitude_g magnitude_blue magnitude_red +=============== =========== ======== =========== ============== ============= +0.00017862 8.3716516 6.857487 12.03939 12.56381 11.36282 +0.00031119 58.557541 2.518501 12.37415 12.81931 11.75581 +0.00058982 5.3805172 6.660454 12.95553 13.58890 12.19339 +0.00106097 -55.64880 7.893578 13.52746 14.63624 12.45566 +0.00113075 3.0206027 4.043843 12.74578 13.15043 12.17218 +... ... ... ... ... ... +=============== =========== ======== =========== ============== ============= + +Here is the meaning of the columns: + +0. ``right_ascension``: horizontal position of the star along the celestial equator (degrees) +#. ``declination``: vertical position of the star above/below celestial equator (degrees) +#. ``parallax``: amount the star's position in the sky changes throughout the year +#. ``magnitude_g``: how bright the star appears in our sky (inverse Logarithmic scale) +#. ``magnitude_blue``: blue-component of the color of the star +#. ``magnitude_red``: red-component of the color of the star We can use one of NumPy's functions to load the entire file into memory:: import numpy as np # we usually use `np` as shorthand - # We set `skip_header=1` to skip the first line containing the description of the values. + # We set `skip_header=1` to skip the first line containing the column names. data = np.genfromtxt("../resources/data/numpy/local_stars.csv", delimiter=",", skip_header=1) print(data) The ``data`` variable is a NumPy array:: print(type(data)) - What is an array? ----------------- @@ -91,14 +120,7 @@ Let's see how many stars we have. The ``.shape`` property of an array contains t print(data.shape) -We have information on more than a million stars, and for each star we have 6 values, corresponding to (see ``names`` above): - -0. ``right_ascension``: horizontal position of the star along the celestial equator (degrees) -#. ``declination``: vertical position of the star above/below celestial equator (degrees) -#. ``parallax``: amount the star's position in the sky changes throughout the year -#. ``magnitude_g``: how bright the star appears in our sky (inverse Logarithmic scale) -#. ``magnitude_red``: red-component of the color of the star -#. ``magnitude_blue``: blue-component of the color of the star +We have information on more than a million stars, and for each star we have 6 values, corresponding to 6 columns listed above. Working with arrays @@ -510,3 +532,19 @@ Exercises 2 The Large Magallanic Cloud is roughly 158000 light-years away. This means it does not lie in our galaxy, but must be a galaxy of its own. + + +Where to go from here +--------------------- +To get started using NumPy for your own research projects, we recommend diving into a more detailed tutorial such as the ones offered by the official NumPy website: https://numpy.org/learn. + + +About the data +-------------- +The data for this exercise is a combination of the `Gaia `_ and `SIMBAD `_ datasets. +The Gaia satellite did not image the brightest stars, the most famous ones, because their brightness overwhelms her sensors. +Hence, the 50 brightest stars our dataset were not from Gaia, but were instead taken from the SIMBAD database. +Those are the stars we look at during the Big Dipper demo. + +For the Large Magallanic Cloud exercise, we cherry picked the stars in :download:`../resources/data/numpy/lmc_stars.csv` to be perfectly on the main sequence and thus ensuring that you get a decent estimate of the distance. +In reality it's quite an ordeal to choose the proper stars to base your distance estimate on. From 543ce25aff078dc66fedf5aba4669d57b286304b Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Mon, 12 Oct 2026 12:58:08 +0300 Subject: [PATCH 19/31] Formatting fix --- content/numpy-stars.rst | 1 + 1 file changed, 1 insertion(+) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index a870d33f..a653cbbf 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -13,6 +13,7 @@ NumPy .. objectives:: This is not a complete NumPy tutorial. Instead, this lesson strives to teach you: + - How to use basic NumPy: arrays, selecting data, performing mathematical operations - A sense of how NumPy code looks: vectorized code. - To understand enough of NumPy to search for answers to the rest of your questions. ;) From c008a2801f01ce5f6b75e985611dd1451354bd50 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Tue, 13 Oct 2026 08:41:03 +0300 Subject: [PATCH 20/31] decided to reorder some things based on practicing given the lesson --- content/numpy-stars.rst | 51 ++++++++++++++++++++--------------------- 1 file changed, 25 insertions(+), 26 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index a653cbbf..7cabbf0f 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -66,28 +66,6 @@ The first line of that file is the "header" indicating the meaning of each numbe 0.0011307510147073,3.0206027981262924,4.043843423571794,12.745788,13.150434,12.17218 ... -Here is the data, formatted as a table: - -=============== =========== ======== =========== ============== ============= -right_ascension declination parallax magnitude_g magnitude_blue magnitude_red -=============== =========== ======== =========== ============== ============= -0.00017862 8.3716516 6.857487 12.03939 12.56381 11.36282 -0.00031119 58.557541 2.518501 12.37415 12.81931 11.75581 -0.00058982 5.3805172 6.660454 12.95553 13.58890 12.19339 -0.00106097 -55.64880 7.893578 13.52746 14.63624 12.45566 -0.00113075 3.0206027 4.043843 12.74578 13.15043 12.17218 -... ... ... ... ... ... -=============== =========== ======== =========== ============== ============= - -Here is the meaning of the columns: - -0. ``right_ascension``: horizontal position of the star along the celestial equator (degrees) -#. ``declination``: vertical position of the star above/below celestial equator (degrees) -#. ``parallax``: amount the star's position in the sky changes throughout the year -#. ``magnitude_g``: how bright the star appears in our sky (inverse Logarithmic scale) -#. ``magnitude_blue``: blue-component of the color of the star -#. ``magnitude_red``: red-component of the color of the star - We can use one of NumPy's functions to load the entire file into memory:: import numpy as np # we usually use `np` as shorthand @@ -127,6 +105,28 @@ We have information on more than a million stars, and for each star we have 6 va Working with arrays ------------------- +Here is the data, formatted as a table: + +=============== =========== ======== =========== ============== ============= +right_ascension declination parallax magnitude_g magnitude_blue magnitude_red +=============== =========== ======== =========== ============== ============= +0.00017862 8.3716516 6.857487 12.03939 12.56381 11.36282 +0.00031119 58.557541 2.518501 12.37415 12.81931 11.75581 +0.00058982 5.3805172 6.660454 12.95553 13.58890 12.19339 +0.00106097 -55.64880 7.893578 13.52746 14.63624 12.45566 +0.00113075 3.0206027 4.043843 12.74578 13.15043 12.17218 +... ... ... ... ... ... +=============== =========== ======== =========== ============== ============= + +Here is the meaning of the columns: + +0. ``right_ascension``: horizontal position of the star along the celestial equator (degrees) +#. ``declination``: vertical position of the star above/below celestial equator (degrees) +#. ``parallax``: amount the star's position in the sky changes throughout the year +#. ``magnitude_g``: how bright the star appears in our sky (inverse Logarithmic scale) +#. ``magnitude_blue``: blue-component of the color of the star +#. ``magnitude_red``: red-component of the color of the star + Let's compute the distance to some well known stars in our night's sky. For that, we need to extract the parallax measurements from the ``data`` array, which are in the column with index ``2``. The way the array is laid out in memory makes it very fast to select portions of the data. @@ -154,14 +154,13 @@ Common mathematical operations include: ``+`` (:data:`numpy.add`), ``-`` (:data: Let's examine the closest star in the dataset:: - distance_to_closest_star = distance_ly.min() # returns the minimum value - index_of_closest_star = distance_ly.argmin() # returns the index of the minimum value + distance_to_closest_star = np.min(distance_ly) # returns the minimum value + index_of_closest_star = np.argmin(distance_ly) # returns the index of the minimum value print(distance_to_closest_star, index_of_closest_star) The closest star is `Proxima Centauri `__ at 4.2465 light-years. To see all the information we have on it, we can select the entire row using the index we just obtained:: - print(names) # the column names for reference print(data[index_of_closest_star, :]) # select the row corresponding to Proxima Centauri NumPy has many other ways to select data. @@ -353,7 +352,7 @@ We will do this using a ``for`` loop and use a typical pattern for creating NumP bin_magnitudes = list() # Loop over all the bins and compute the median color/magnitude. - for bin_index in np.arange(n_bins): + for bin_index in range(n_bins): bin_color = np.median(color[bin_assignment == bin_index]) bin_magnitude = np.median(magnitude[bin_assignment == bin_index]) bin_colors.append(bin_color) From 36fa603debfde85fa7fb787a14e2afa3bd55cd20 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Tue, 13 Oct 2026 09:13:14 +0300 Subject: [PATCH 21/31] Better Big Dipper image --- content/img/numpy/Big_Dipper.jpg | Bin 267280 -> 0 bytes content/img/numpy/big_dipper.svg | 8882 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Among the things it measured was the "parallax" of each object, which is how much it seems to "move" in relationship to very distant stars as the sattelite traveled around the sun and observed it from different angles. We can use the amount of parallax to determine how far away a star is. -.. image:: img/numpy/parsec.png +.. figure:: img/numpy/parsec.png :width: 300px :align: center + Visual explanation of the parallax effect. Image `retrieved from Wikipedia `__. Public domain. + A sample of the Gaia data can be found in :download:`../resources/data/numpy/local_stars.csv`. This is a *comma separated value (CSV)* file, where each line contains 6 numbers describing a single object. The first line of that file is the "header" indicating the meaning of each number:: @@ -93,7 +95,7 @@ Let's see how many stars we have:: We have information on more than a million stars, and for each star we have 6 values, corresponding to (see ``names`` above): -0. ``right_ascention``: horizontal position of the star along the celestial equator (degrees) +0. ``right_ascension``: horizontal position of the star along the celestial equator (degrees) #. ``declination``: vertical position of the star above/below celestial equator (degrees) #. ``parallax``: amount the star's position in the sky changes throughout the year #. ``magnitude_g``: how bright the star appears in our sky (inverse Logarithmic scale) @@ -154,30 +156,35 @@ We can select multiple rows by indexing the array with such a list:: closest_stars = data[closest_indices, :] print(distance_ly[closest_indices]) -Another powerful indexing technique is to create an index based on boolean values. +Can we find the `Big Dipper `__? + +.. figure:: img/numpy/Big_Dipper.jpg + :width: 300px + :align: center + + The Big Dipper as seen from Fujian. Image `retrieved from Wikipedia `__. CC-BY + + +It lies within a section of the sky with a `right ascension `__ from 160 to 210 degrees, and a `declination `__ from 45 to 65 degrees:: + + right_ascension = data[:, 0] + declination = data[:, 1] + +To select elements from an array based on some condition, we can create an index based on boolean values. Recall that in Python, comparison operations (``==``, ``<``, ``<=``, ``>``, ``>=``) produce a boolean (``True``/``False``) value. See what happens when you apply them to an array:: - closer_than_15_ly = distance_ly < 15 - print(closer_than_15_ly.shape) - print(closer_than_15_ly[:10]) + print(right_ascension > 160) It produces a new array, of the same dimensions as the original array, but filled with ``True``/``False`` values indicating for each element whether the comparison holds true or not. -We call this a "boolean mask" and we can use it to index another array:: +We call this a "boolean mask". +We can combine multiple masks using Python's boolean "and" operator ``&``:: - print(distance_ly[closer_than_15_ly]) - print(data[closer_than_15_ly, :].shape) + boolean_mask = (right_ascension > 160) & (right_ascension < 210) & (declination > 45) & (declination < 65) -Can we find the `Big Dipper `__? -It lies within a section of the sky with a right ascention from 160 to 210 degrees, and a declination from 45 to 65 degrees:: +Once you have a boolean mask, you can apply it to any array as long at the dimension along which you are selecting has the same number of elements as the mask:: - right_ascention = data[:, 0] - declination = data[:, 1] - boolean_mask = ( - (right_ascention > 160) & (right_ascention < 210) & - (declination > 45) & (declination < 65) - ) - sky_section = data[boolean_mask] + sky_section = data[boolean_mask, :] print(sky_section.shape) In our dataset, there are more than 15k stars in our chosen section of sky. @@ -186,26 +193,30 @@ Column 3, named `magnitude_g` denotes bright the star looks as seen by the Gaia The `magnitude scale `__ works "in reverse", such that very bright stars have a magnitude of 1 and weaker stars have a larger magnitude. .. code:: + + # Select the 7 brightest stars in our section of sky. order_by_magnitude = np.argsort(sky_section[:, 3]) sky_section = sky_section[order_by_magnitude, :] big_dipper = sky_section[:7] - # Make a figure with the position of the stars. We will cover this in a later lesson. + # Make a figure with the position of the stars in the sky. You can take this at face + # value for now. We will cover how to make figures in a later lesson. + import matplotlib.pyplt as plt fig, ax = plt.subplots() ax.scatter(big_dipper[:, 0], big_dipper[:, 1]) - ax.xaxis.set_inverted(True) + ax.xaxis.set_inverted(True) # right ascension goes from right-to-left ax.set_aspect("equal") - ax.set_xlabel("right ascention (degrees)") + ax.set_xlabel("right ascension (degrees)") ax.set_ylabel("declination (degrees)") +Clever and efficient use of these operations is a key to NumPy's speed: you should try to cleverly use these selectors (written in C) to extract data to be used with other NumPy functions written in C or Fortran. +This will give you the benefits of Python with most of the speed of C. + .. seealso:: :ref:`Numpy basic indexing docs ` -Clever and efficient use of these operations is a key to NumPy's speed: you should try to cleverly use these selectors (written in C) to extract data to be used with other NumPy functions written in C or Fortran. -This will give you the benefits of Python with most of the speed of C. - Exercises 1 ----------- @@ -222,43 +233,90 @@ Exercises 1 #. How many stars are there in the dataset that are even brighter than the brightest star in the Big Dipper (`Alioth `__)? + +Part 2: Estimating the distance of really far away stars +-------------------------------------------------------- + +When stars are very far away, the parallax is so small even the Gaia satellite cannot reliably measure it. +Still, we can estimate roughly how far away they are based on a striking relationship between the brightness and the color of a star. +A `Hertzsprung-Russel diagram `__ illustrates this. +Let's make one! + +In our data, column 3 contains the measurements of the apparent brightness of each star according to Gaia's sensors. +For the Hertzsprung-Russel diagram, we need to estimate the intrinsic brightness (the luminosity) of the stars, compensating for the fact that the further a star is, the dimmer it appears to us. +We will use the formula given in the `official Gaia paper `__ for this:: + + magnitude_g = data[:, 3] # apparent magnitude + magnitude = magnitude_g + 5 * np.log10(parallax) - 10 # true magnitude + +In our diagram, we want to show the color of a star on a scale from "very blue" to "very red", with yellow stars in between. +We can compute this by taking the difference between the blue and red components of its color:: + + magnitude_red = data[:, 4] + magnitude_blue = data[:, 5] + color = magnitude_blue - magnitude_red + +Now we have everything we need to make the plot. Again, just take this plotting code at face value for now:: + + import matplotlib.pyplot as plt + plt.figure(figsize=(6, 8)) + plt.scatter( + color, magnitude, s=1, c=distance_ly, cmap="plasma", alpha=0.1 + ) + cb = plt.colorbar(label="Distance (light years)", fraction=0.05, aspect=40) + cb.solids.set(alpha=1) # make the colors in the colorbar not be transparent + plt.gca().invert_yaxis() # put bright stars on top, dim stars at the bottom + plt.title("Hertzsprung-Russell diagram") + plt.xlabel("Color (Blue ⇨ Red)") + plt.ylabel("Magnitude") + plt.tight_layout() + +Other than the main sequence stars, you can see the red giants branching off at the top, bright but red, and the white dwarfs at the bottom, very dim but white. + +Based on this chart, we can derive a formula to translate the color of a main sequence star (so not a red giant or white dwarf) to its true brightness. +Then, we can compare that true brightness with its apparent brightness to deduce how much dimmer it appears to us than it actually is, hence how far away it must be. + + Creating arrays --------------- -Now we know how far stars are from us, but how far are stars from each other? -This is easier to answer if we transform the coordinates from Gaia-CRF (ra, dec, distance) in degrees -> Cartesian (x, y, z) in light-years. -Let's create a new 2D array of shape ``(n_stars, 3)`` with the Cartesian coordinates of each star, measured in light-years, with us at the center. - -We can compute each coordinate using the trigonometry functions `np.sin` and `np.cos`. -These operate in radians, so we use `np.deg2rad()` to convert degrees to radians:: - - x = distance_ly * np.cos(np.deg2rad(declination)) * np.cos(np.deg2rad(right_ascention)) - y = distance_ly * np.cos(np.deg2rad(declination)) * np.sin(np.deg2rad(right_ascention)) - z = distance_ly * np.sin(np.deg2rad(declination)) - -One of the many ways to create an array is to start with a Python list. -Now currently have three 1D arrays. -To create a 2D array out of them, we packing them into a Python list and then convert this list to an array:: - - xyz_list = [x, y, z] - xyz = np.array(xyz_list) - print(xyz.shape) - -This gives us an array with 3 rows and each star being a column. -To swap rows and columns is called the `Transpose `__ and is such a common operation, especially when doing linear algebra, that NumPy has a shorthand for it:: - - xyz = xyz.T # transpose the 2D array - print(xyz.shape) - -# Question: Alkaid and Mizar form the tail end of the "big dipper". -# How far are these stars from one another? -# * Dubhe -# * Mizar Megrez -# / * * -# * Alkaid / Alioth * Merak -# \ / * -# \ / Phecda -# \ / -# \ / -# * Sol -# +The first step is to extract a bunch of ``(color, magnitude)`` pairs along the main sequence, free from all the outliers. +The strategy is to divide the color spectrum into 100 bins and take the median magnitude in each bin. +With the vast majority of the stars being main sequence stars, the median should be very representative of the general curve of the main sequence. + +There are various ways to create an array with equally spaced numbers. +For example, there is the NumPy equivalent of Python's :func:`range` function, :func:`numpy.arange`:: + + bins = np.arange(0.5, 4, step=0.04) + +But for our purposes, :func:`numpy.linspace` is even better:: + + bins = np.linspace(0.5, 4, num=100) # 100 numbers evenly spread between 0.5 and 4 + +We could assign stars to their respective color bins using boolean masking, but NumPy offers the :func:`numpy.digitize` function especially for this purpose:: + + bin_assignment = np.digitize(color, bins) + print(bin_assignment) + +The function gives for each element of the ``color`` array (as an integer index), the index of the bin it belongs to. +Since each element of the ``color`` array is the color of a star, we have now effectively assigned each star to a bin. +Now we can compute for each bin, the median brightness and color of all the stars in that bin. +We will do this using a ``for`` loop and use a typical pattern for creating NumPy arrays: we first collect the values into a Python list and then convert that list into a NumPy array:: + + bin_colors = list() # we collect the median colors here + bin_magnitudes = list() # we collect the median magnitudes here + for star_indices in bin_assignment: + bin_color = np.median(color[star_indices]) + bin_magnitude = np.median(magnitude[star_indices]) + bin_colors.append(bin_color) + bin_magnitudes.append(bin_magnitude) + # Convert the Python lists to NumPy arrays. + bin_colors = np.array(bin_colors) + bin_magnitudes = np.array(bin_magnitudes) + +.. note:: + Appending values to a Python list is very fast, but appending values to a NumPy array is very slow. This is why we first collect the values in a Python list and then convert it to a NumPy array. + +.. seealso:: + + `Numpy array creation docs `_ From 8880bc3fe818f158b028f7e09294a905439cb122 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Mon, 29 Jun 2026 23:43:05 +0300 Subject: [PATCH 04/31] Continue working on NumPy lesson --- content/numpy-stars.rst | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index e1feeb7b..3a4dd5c8 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -299,8 +299,8 @@ We could assign stars to their respective color bins using boolean masking, but print(bin_assignment) The function gives for each element of the ``color`` array (as an integer index), the index of the bin it belongs to. -Since each element of the ``color`` array is the color of a star, we have now effectively assigned each star to a bin. -Now we can compute for each bin, the median brightness and color of all the stars in that bin. +Since each element of the ``color`` array is the color of a star, we have effectively assigned each star to a bin. +Now, we can compute for each bin, the median brightness and color of all the stars in that bin. We will do this using a ``for`` loop and use a typical pattern for creating NumPy arrays: we first collect the values into a Python list and then convert that list into a NumPy array:: bin_colors = list() # we collect the median colors here From 8066ece34a2c4dfd98d916a8d1e154b83735d40c Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Thu, 2 Jul 2026 10:11:38 +0300 Subject: [PATCH 05/31] Continue working on NumPy lesson --- content/numpy-stars.rst | 135 ++++++++++++++++++++++++++++++++++++++-- 1 file changed, 129 insertions(+), 6 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index 3a4dd5c8..1d8d627b 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -291,7 +291,8 @@ For example, there is the NumPy equivalent of Python's :func:`range` function, : But for our purposes, :func:`numpy.linspace` is even better:: - bins = np.linspace(0.5, 4, num=100) # 100 numbers evenly spread between 0.5 and 4 + n_bins = 100 + bins = np.linspace(0.5, 4, num=n_bins) # 100 numbers evenly spread between 0.5 and 4 We could assign stars to their respective color bins using boolean masking, but NumPy offers the :func:`numpy.digitize` function especially for this purpose:: @@ -303,13 +304,17 @@ Since each element of the ``color`` array is the color of a star, we have effect Now, we can compute for each bin, the median brightness and color of all the stars in that bin. We will do this using a ``for`` loop and use a typical pattern for creating NumPy arrays: we first collect the values into a Python list and then convert that list into a NumPy array:: - bin_colors = list() # we collect the median colors here - bin_magnitudes = list() # we collect the median magnitudes here - for star_indices in bin_assignment: - bin_color = np.median(color[star_indices]) - bin_magnitude = np.median(magnitude[star_indices]) + # We collect the median colors and magnitudes in these lists. + bin_colors = list() + bin_magnitudes = list() + + # Loop over all the bins and compute the median color/magnitude. + for bin_index in np.arange(n_bins): + bin_color = np.median(color[bin_assignment == bin_index]) + bin_magnitude = np.median(magnitude[bin_assignment == bin_index]) bin_colors.append(bin_color) bin_magnitudes.append(bin_magnitude) + # Convert the Python lists to NumPy arrays. bin_colors = np.array(bin_colors) bin_magnitudes = np.array(bin_magnitudes) @@ -320,3 +325,121 @@ We will do this using a ``for`` loop and use a typical pattern for creating NumP .. seealso:: `Numpy array creation docs `_ + + +Doing linear algebra +-------------------- + +If we fit a curve through the ``(color, magnitude)`` combinations, we obtain a function to predict a star's magnitude based on its color. +We can do this with a bit of linear algebra. +To get a feel for how the curve should look, let's draw the ``(color, magnitude)`` combinations on top of the Hertzsprung-Russel diagram (assuming it is still open, re-create it if it isn't):: + + plt.plot(bin_colors, bin_magnitudes, color="green", linewidth=2) + +The curve is not a straight line, but something like a fifth order polynomial should be pretty good fit: + +.. math:: + :name: A fifth order polynonial function + + \text{magnitude} = \beta_5\,\text{color}^5 + \beta_4\,\text{color}^4 + \beta_2\,\text{color}^3 + \beta_2\,\text{color}^2 + \beta_1\,\text{color} + \beta_0 + +"Fitting the curve" now means choosing the optimal :math:`\beta` values so that when we input the ``color``, we get a good prediction of ``magnitude``. +A typical way to do this is a machine learning technique called `ordinary least squares (OLS) `__, where we collect everything we know in a matrix ``X`` and everything we wish to predict in a matrix ``Y`` and compute: + +.. math:: + :name: The ordinary least squares function + + \beta = (X^T X)^{-1} X^T Y + +The resulting ``beta`` vector contains the optimal linear combination (in the least-squares sense) of the columns of ``X`` to predict the columns of ``Y``. +So, we create a 2D array ``X`` where the columns are ``bin_colors`` raised to different powers and set ``Y`` to ``bin_magnitude``. +The code below does this, while demonstrating how to create a 2D array by gluing 1D arrays together, and how to generate an array containing only ``1``'s:: + + X = np.column_stack(( + np.ones_like(bin_colors), # bin_colors ** 0 + bin_colors, # bin_colors ** 1 + bin_colors ** 2, + bin_colors ** 3, + bin_colors ** 4, + bin_colors ** 5, + )) + Y = bin_magnitudes + +To compute the OLS formula, we need a bunch of linear algebra functionality: + +* matrix multiplication, which is written as ``@`` in Python +* matrix transpose, which NumPy arrays support through their property ``.T`` +* matrix inversion, which is done through :func:`numpy.linalg.inv` + +Using NumPy, the Python code can look very much like the math formula:: + + beta = np.linalg.inv(X.T @ X) @ X.T @ Y + print(beta) + + +Creating and documenting functions in the scientific computing style +-------------------------------------------------------------------- + +Now that we have ``beta``, we can use formula (1) to predict a star's magnitude given its color. +Let's wrap the formula in a function, so we can give it a name and write documentation on how to use it. +NumPy comes with a style guide for writing docstrings for functions called `NumPyDoc `__:: + + def predict_magnitude(color, beta): + """Predict a main sequence star's inherent magnitude given its color. + + Parameters + ---------- + color : float | array of float, shape (n_stars,) + The color of the star(s), computed as red - blue. + beta: array of float, shape (6,) + The regression weights. + + Returns + ------- + magnitude : float + The estimated intrinsic magnitude of the star(s). + """ + return ( + beta[5] * color ** 5 + beta[4] * color ** 4 + + beta[3] * color ** 3 + + beta[2] * color ** 2 + + beta[1] * color + + beta[0] + + ) + +Let's use our ``predict_magnitude`` function to create a function that predicts the distance of a star given its color and apparent magnitude.:: + + def predict_distance(magnitude_g, magnitude_red, magnitude_blue, beta): + """Predict a main sequence star's distance given its apparent magnitude and color. + + Parameters + ---------- + magintude_g : float | array of float, shape (n_stars,) + The apparent magnitude of the star(s). + magnitude_red : float | array of float, shape (n_stars,) + The red component of the color of the star(s). + magnitude_blue : float | array of float, shape (n_stars,) + The blue component of the color of the star(s). + beta: array of float, shape (6,) + The regression weights. + + Returns + ------- + distance : float + The estimated distance in light-years. + """ + color = magnitude_red - magnitude_blue + predicted_magnitude = predict_magnitude(color, beta) + # Inverse of the formula we used to derive intrinsic magnitude from apparent magnitude. + predicted_distance_parsecs = np.pow(10, ((magnitude_g - predicted_magnitude) + 10) / 5) + return predicted_distance_parsecs * 3.26156 + +A sample of some far away main sequence stars in the Gaia dataset can be found in :download:`../resources/data/numpy/far_stars.csv`. +Their parallax values may be unreliable, so let's predict their distance using main sequence fitting:: + + far_stars = np.genfromtxt("../resources/data/numpy/far_stars.csv", delimiter=",", skip_header=1) + print(far_stars.shape) + + far_stars_distance = predict_distance(far_stars[:, 3], far_stars[:, 4], far_stars[:, 5], beta) + print(far_stars_distance) From 24bec23b91440cd69f274c3e8fc86feb1af7771a Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Thu, 2 Jul 2026 10:13:01 +0300 Subject: [PATCH 06/31] Continue working on NumPy lesson --- content/numpy-stars.rst | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index 1d8d627b..f14b1d22 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -429,11 +429,11 @@ Let's use our ``predict_magnitude`` function to create a function that predicts distance : float The estimated distance in light-years. """ - color = magnitude_red - magnitude_blue - predicted_magnitude = predict_magnitude(color, beta) - # Inverse of the formula we used to derive intrinsic magnitude from apparent magnitude. - predicted_distance_parsecs = np.pow(10, ((magnitude_g - predicted_magnitude) + 10) / 5) - return predicted_distance_parsecs * 3.26156 + color = magnitude_red - magnitude_blue + predicted_magnitude = predict_magnitude(color, beta) + # Inverse of the formula we used to derive intrinsic magnitude from apparent magnitude. + predicted_distance_parsecs = np.pow(10, ((magnitude_g - predicted_magnitude) + 10) / 5) + return predicted_distance_parsecs * 3.26156 A sample of some far away main sequence stars in the Gaia dataset can be found in :download:`../resources/data/numpy/far_stars.csv`. Their parallax values may be unreliable, so let's predict their distance using main sequence fitting:: From 0e635bb598f3d521f1736cab6cdebed3ba3dda38 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Sat, 4 Jul 2026 14:24:00 +0300 Subject: [PATCH 07/31] fix function --- content/numpy-stars.rst | 64 +++++++++++++++++++++++++++-------------- 1 file changed, 42 insertions(+), 22 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index f14b1d22..c781e37e 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -180,7 +180,12 @@ It produces a new array, of the same dimensions as the original array, but fille We call this a "boolean mask". We can combine multiple masks using Python's boolean "and" operator ``&``:: - boolean_mask = (right_ascension > 160) & (right_ascension < 210) & (declination > 45) & (declination < 65) + boolean_mask = ( + (right_ascension > 160) & + (right_ascension < 210) & + (declination > 45) & + (declination < 65) + ) Once you have a boolean mask, you can apply it to any array as long at the dimension along which you are selecting has the same number of elements as the mask:: @@ -201,7 +206,7 @@ The `magnitude scale `__ wo # Make a figure with the position of the stars in the sky. You can take this at face # value for now. We will cover how to make figures in a later lesson. - import matplotlib.pyplt as plt + import matplotlib.pyplot as plt fig, ax = plt.subplots() ax.scatter(big_dipper[:, 0], big_dipper[:, 1]) ax.xaxis.set_inverted(True) # right ascension goes from right-to-left @@ -287,12 +292,12 @@ With the vast majority of the stars being main sequence stars, the median should There are various ways to create an array with equally spaced numbers. For example, there is the NumPy equivalent of Python's :func:`range` function, :func:`numpy.arange`:: - bins = np.arange(0.5, 4, step=0.04) + bins = np.arange(0.1, 5, step=0.04) But for our purposes, :func:`numpy.linspace` is even better:: n_bins = 100 - bins = np.linspace(0.5, 4, num=n_bins) # 100 numbers evenly spread between 0.5 and 4 + bins = np.linspace(0.1, 5, num=n_bins) # 100 numbers evenly spread between 0.1 and 4 We could assign stars to their respective color bins using boolean masking, but NumPy offers the :func:`numpy.digitize` function especially for this purpose:: @@ -319,6 +324,11 @@ We will do this using a ``for`` loop and use a typical pattern for creating NumP bin_colors = np.array(bin_colors) bin_magnitudes = np.array(bin_magnitudes) + # The white dwarf stars with a color < 0.1 are all lumped in the first bin, + # so that bin is unreliable. + bin_colors = bin_colors[1:] + bin_magnitudes = bin_magnitudes[1:] + .. note:: Appending values to a Python list is very fast, but appending values to a NumPy array is very slow. This is why we first collect the values in a Python list and then convert it to a NumPy array. @@ -390,7 +400,7 @@ NumPy comes with a style guide for writing docstrings for functions called `NumP Parameters ---------- color : float | array of float, shape (n_stars,) - The color of the star(s), computed as red - blue. + The color of the star(s), computed as magnitude_blue - magnitude_red. beta: array of float, shape (6,) The regression weights. @@ -400,27 +410,35 @@ NumPy comes with a style guide for writing docstrings for functions called `NumP The estimated intrinsic magnitude of the star(s). """ return ( - beta[5] * color ** 5 + beta[5] * color ** 5 + beta[4] * color ** 4 + beta[3] * color ** 3 + beta[2] * color ** 2 + beta[1] * color + - beta[0] + + beta[0] ) -Let's use our ``predict_magnitude`` function to create a function that predicts the distance of a star given its color and apparent magnitude.:: +To do the final estimation of a star's distance, we can compare our estimate of the intrinsic magnitude of a star based on its color, with the apparent magnitude of the star on our sensors. +The further away the star is, the dimmer it appears to us. +Recall this formula we used earlier:: + + magnitude = magnitude_g + 5 * np.log10(parallax) - 10 + +We can solve for ``parallax``, which gives us:: - def predict_distance(magnitude_g, magnitude_red, magnitude_blue, beta): + estimated_parallax = np.pow(10, (magnitude - magnitude_g + 10) / 5) + +That makes our final function that predicts the distance of a star given its color and apparent magnitude:: + + def predict_distance(magnitude_g, color, beta): """Predict a main sequence star's distance given its apparent magnitude and color. Parameters ---------- magintude_g : float | array of float, shape (n_stars,) The apparent magnitude of the star(s). - magnitude_red : float | array of float, shape (n_stars,) - The red component of the color of the star(s). - magnitude_blue : float | array of float, shape (n_stars,) - The blue component of the color of the star(s). + color : float | array of float, shape (n_stars,) + The color of the star(s), computed as magnitude_blue - magnitude_red. beta: array of float, shape (6,) The regression weights. @@ -429,17 +447,19 @@ Let's use our ``predict_magnitude`` function to create a function that predicts distance : float The estimated distance in light-years. """ - color = magnitude_red - magnitude_blue - predicted_magnitude = predict_magnitude(color, beta) - # Inverse of the formula we used to derive intrinsic magnitude from apparent magnitude. - predicted_distance_parsecs = np.pow(10, ((magnitude_g - predicted_magnitude) + 10) / 5) - return predicted_distance_parsecs * 3.26156 + magnitude = predict_magnitude(color, beta) + parallax = np.pow(10, (magnitude - magnitude_g + 10) / 5) + distance_parsecs = 1 / (parallax / 1000) + return distance_parsecs * 3.26156 # convert to light-years -A sample of some far away main sequence stars in the Gaia dataset can be found in :download:`../resources/data/numpy/far_stars.csv`. -Their parallax values may be unreliable, so let's predict their distance using main sequence fitting:: +Some far away main sequence stars from the Gaia dataset can be found in :download:`../resources/data/numpy/far_stars.csv`. +Let's predict their distance using main sequence fitting, and compare that with the distance we obtain from their parallax values:: far_stars = np.genfromtxt("../resources/data/numpy/far_stars.csv", delimiter=",", skip_header=1) print(far_stars.shape) - far_stars_distance = predict_distance(far_stars[:, 3], far_stars[:, 4], far_stars[:, 5], beta) - print(far_stars_distance) + distance_from_color = predict_distance(far_stars[:, 3], far_stars[:, 4], far_stars[:, 5], beta) + distance_from_parallax = 3.28156 * (1 / (far_stars[:, 2] / 1000)) + + print(distance_from_color[:10]) + print(distance_from_parallax[:10]) From 27a4556bc54b6d2c3e805a21bf51085511f6a359 Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Sat, 4 Jul 2026 15:41:38 +0300 Subject: [PATCH 08/31] Finish first version of the NumPy lesson --- content/numpy-stars.rst | 125 ++++++++++++++++++++++++---------------- 1 file changed, 76 insertions(+), 49 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index c781e37e..c55c4e9a 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -162,7 +162,7 @@ Can we find the `Big Dipper `__? :width: 300px :align: center - The Big Dipper as seen from Fujian. Image `retrieved from Wikipedia `__. CC-BY + The Big Dipper as seen from Fujian. Image `retrieved from Wikipedia `__. CC BY license. It lies within a section of the sky with a `right ascension `__ from 160 to 210 degrees, and a `declination `__ from 45 to 65 degrees:: @@ -235,15 +235,32 @@ Exercises 1 It is the second closest star to us. What is the distance from us to Alpha Centauri AB? #. What is the distance from us to the brightest star in the dataset (`Sirius `__)? + Remember that a smaller magnitude means a brighter star. + You can use :func:`numpy.argmin` to compute the index of the minimum value in an array. #. How many stars are there in the dataset that are even brighter than the brightest star in the Big Dipper (`Alioth `__)? + You can use :func:`numpy.min` to compute the minimum value in an array. +.. solution:: Solution Numpy-1 + + #. Row index: ``alpha_centauri_ind = np.argsort(distance_ly)[1]``. Its distance is ``distance_ly[alpha_centauri_ind]`` = 4.395 light-years. + #. Brightest star index: ``brightest_ind = np.argsort(data[:, 3])[0]``. Its distance is ``distance_ly[brightest_ind]`` = 8.601 light-years. + #. The brightest star in the Big Dipper has a magnitude of ``np.min(big_dipper[:, 3])`` = 1.77. + There are ``data[data[:, 3] < np.min(big_dipper[:, 3])].shape`` = 28 stars in the dataset that are even brighter. Part 2: Estimating the distance of really far away stars -------------------------------------------------------- -When stars are very far away, the parallax is so small even the Gaia satellite cannot reliably measure it. -Still, we can estimate roughly how far away they are based on a striking relationship between the brightness and the color of a star. +On the southern hemisphere, you can see a large cluster of stars, 12 times wider than the moon: + +.. figure:: img/numpy/Large_Magellanic_Cloud.jpg + :width: 500px + :align: center + + The Large Magallanic Cloud. Image created by Martin Bernard, `retrieved from Wikipedia `__. CC BY-SA license. + +It is so far away, the parallax is so small that even the Gaia satellite cannot reliably measure it. +Still, we can estimate roughly how far away it is, based on a striking relationship between the brightness and the color of a star. A `Hertzsprung-Russel diagram `__ illustrates this. Let's make one! @@ -257,8 +274,8 @@ We will use the formula given in the `official Gaia paper `__ galaxy is about 87400 light-years across, what does the distance to the Large Magallanic Cloud tell us? + +.. solution:: Solution Numpy-2 + + .. code:: + + def predict_distance(magnitude_g, color, beta): + """Predict a main sequence star's distance given its apparent magnitude and color. + + Parameters + ---------- + magnitude_g : float | array of float, shape (n_stars,) + The apparent magnitude of the star(s). + color : float | array of float, shape (n_stars,) + The color of the star(s), computed as magnitude_blue - magnitude_red. + beta: array of float, shape (6,) + The regression weights. + + Returns + ------- + distance : float + The estimated distance in light-years. + """ + magnitude = predict_magnitude(color, beta) + parallax = np.pow(10, (magnitude - magnitude_g + 10) / 5) + distance_parsecs = 1 / (parallax / 1000) + return distance_parsecs * 3.26156 # convert to light-years + + lmc = np.genfromtxt("../resources/data/numpy/lmc_stars.csv", delimiter=",", skip_header=1) + distance = predict_distance(lmc[:, 3], lmc[:, 4] - lmc[:, 5], beta) + print(distance.mean()) From 498096493f95e428ebc1160de851ef57e030ebbf Mon Sep 17 00:00:00 2001 From: Marijn van Vliet Date: Sun, 5 Jul 2026 11:05:54 +0300 Subject: [PATCH 09/31] Improve the text --- content/numpy-stars.rst | 151 ++++++++++++++++++++++------------------ 1 file changed, 83 insertions(+), 68 deletions(-) diff --git a/content/numpy-stars.rst b/content/numpy-stars.rst index c55c4e9a..a4b8adfb 100644 --- a/content/numpy-stars.rst +++ b/content/numpy-stars.rst @@ -47,7 +47,7 @@ We can use the amount of parallax to determine how far away a star is. :width: 300px :align: center - Visual explanation of the parallax effect. Image `retrieved from Wikipedia `__. Public domain. + Visual explanation of the parallax effect. Image `retrieved from WikiMedia `__. Public domain. A sample of the Gaia data can be found in :download:`../resources/data/numpy/local_stars.csv`. This is a *comma separated value (CSV)* file, where each line contains 6 numbers describing a single object. @@ -56,7 +56,7 @@ The first line of that file is the "header" indicating the meaning of each numbe with open("../resources/data/numpy/local_stars.csv") as f: # The file handle `f` acts as a generator that yields lines of text. first_line = next(f) # take a single item from the generator - names = first_line.split(",") # split on commas to get a list of individual names + names = first_line.strip().split(",") # get a list of individual names print(names) We can use one of NumPy's functions to load the entire file into memory:: @@ -74,22 +74,22 @@ The ``data`` variable is a NumPy array:: What is an array? ----------------- -An array is a "grid" of values, with all the same type, tightly packed into memory side-by-side. -This is why we skipped the first line of the file containing ``str`` type descriptions, so that we were only reading numbers. -In our case, the data type of the array is:: +An array is a collection of values which are all of the same datatype, tightly packed side-by-side in a continuous chunk of memory. +This is why we had to skip the first line of the file, the one containing ``str`` type descriptions, so that we were only reading numbers. +In our case, the datatype of the resulting array is:: print(data.dtype) -Arrays can have any number of dimensions:: +Arrays can have any number of dimensions, ours has:: print(data.ndim) You can think of a 1D array as a *list* and a 2D array as a *table*. -In mathematics, we call a 1D array a *vector*, a 2D array a *matrix*, and an array with 3 or even more dimensions a *tensor*. +Alternatively, in mathematics, we call a 1D array a *vector*, a 2D array a *matrix*, and an array with 3 or even more dimensions a *tensor*. An array can even have zero dimensions if it represents a single *scalar* number. Regardless of how many dimensions the array has, it is always a Python object of type :class:`numpy.ndarray`. -Let's see how many stars we have:: +Let's see how many stars we have. The ``.shape`` property of an array contains the size along each dimension:: print(data.shape) @@ -106,16 +106,17 @@ We have information on more than a million stars, and for each star we have 6 va Working with arrays ------------------- -The way the array is laid out in memory makes it very fast to select portions of the data, by "indexing" or "slicing" the array. -You can index/slice arrays in a similar manner as Python lists, but you specify an index/slice for each dimension. -At the moment, we are only interested in the parallax value of each star, so let's extract that column and store it in a variable of its own:: +Let's compute the distance to some well known stars in our night's sky. +For that, we need to extract the parallax measurements from the ``data`` array, which are in the column with index ``2``. +The way the array is laid out in memory makes it very fast to select portions of the data. +We can index/slice arrays in a similar manner as Python lists using ``[index]``, but we do this separately along each dimension: ``[row_index, col_index]``:: parallax = data[:, 2] # `:` is a shorthand for "select everything along this dimension" print(parallax) Now for some basic math. The parallax values are currently in `milli-arc-seconds `__. -Let's convert that into distance. +We need to convert that into distance. A common distance metric is the `"parsec" `__, which is the distance at which an object has a parallax of exactly one arc-second, which is 3.26156 light-years:: distance_parsecs = 1 / (parallax / 1000) # 1000 milli-arc-seconds in an arc-second @@ -150,7 +151,7 @@ To do this, we must first order the stars by distance:: closest_indices = order_by_distance[:10] # select the first 10 elements print(closest_indices) -We now have a list (technically a 1D numpy.array) of integer indices of the stars closest to us. +We now have a list (technically a 1D :class:`numpy.ndarray`) of integer indices of the stars closest to us. We can select multiple rows by indexing the array with such a list:: closest_stars = data[closest_indices, :] @@ -162,7 +163,7 @@ Can we find the `Big Dipper `__? :width: 300px :align: center - The Big Dipper as seen from Fujian. Image `retrieved from Wikipedia `__. CC BY license. + The Big Dipper as seen from Fujian. Image `retrieved from WikiMedia `__. CC BY license. It lies within a section of the sky with a `right ascension `__ from 160 to 210 degrees, and a `declination `__ from 45 to 65 degrees:: @@ -187,22 +188,22 @@ We can combine multiple masks using Python's boolean "and" operator ``&``:: (declination < 65) ) -Once you have a boolean mask, you can apply it to any array as long at the dimension along which you are selecting has the same number of elements as the mask:: +Once you have a boolean mask, you can apply it to any dimension of the array as long as it has the same number of elements as the mask:: sky_section = data[boolean_mask, :] print(sky_section.shape) In our dataset, there are more than 15k stars in our chosen section of sky. The Big Dipper should consist of the 7 brightest ones. -Column 3, named `magnitude_g` denotes bright the star looks as seen by the Gaia sattelite. +The column with index ``3``, named ``magnitude_g``, denotes how bright the star looks as seen by the Gaia sattelite. The `magnitude scale `__ works "in reverse", such that very bright stars have a magnitude of 1 and weaker stars have a larger magnitude. .. code:: # Select the 7 brightest stars in our section of sky. order_by_magnitude = np.argsort(sky_section[:, 3]) - sky_section = sky_section[order_by_magnitude, :] - big_dipper = sky_section[:7] + brightest_stars = order_by_magnitude[:7] + big_dipper = sky_section[brightest_stars, :] # Make a figure with the position of the stars in the sky. You can take this at face # value for now. We will cover how to make figures in a later lesson. @@ -215,7 +216,7 @@ The `magnitude scale `__ wo ax.set_ylabel("declination (degrees)") -Clever and efficient use of these operations is a key to NumPy's speed: you should try to cleverly use these selectors (written in C) to extract data to be used with other NumPy functions written in C or Fortran. +Clever and efficient use of these indexing and mathematical operations is the key to NumPy's speed: you should try to cleverly use these selectors (written in C) to extract data to be used with other NumPy functions written in C or Fortran. This will give you the benefits of Python with most of the speed of C. .. seealso:: @@ -236,9 +237,7 @@ Exercises 1 What is the distance from us to Alpha Centauri AB? #. What is the distance from us to the brightest star in the dataset (`Sirius `__)? Remember that a smaller magnitude means a brighter star. - You can use :func:`numpy.argmin` to compute the index of the minimum value in an array. #. How many stars are there in the dataset that are even brighter than the brightest star in the Big Dipper (`Alioth `__)? - You can use :func:`numpy.min` to compute the minimum value in an array. .. solution:: Solution Numpy-1 @@ -251,26 +250,28 @@ Exercises 1 Part 2: Estimating the distance of really far away stars -------------------------------------------------------- -On the southern hemisphere, you can see a large cluster of stars, 12 times wider than the moon: +When you are on the southern hemisphere of the Earth, you can see a massive cluster of stars, 20 times wider than the moon: .. figure:: img/numpy/Large_Magellanic_Cloud.jpg :width: 500px :align: center - The Large Magallanic Cloud. Image created by Martin Bernard, `retrieved from Wikipedia `__. CC BY-SA license. + The Large Magallanic Cloud. Photographed and edited by Martin Bernard, `retrieved from WikiMedia `__. CC BY-SA license. -It is so far away, the parallax is so small that even the Gaia satellite cannot reliably measure it. -Still, we can estimate roughly how far away it is, based on a striking relationship between the brightness and the color of a star. +Its parallax is so small that even the Gaia satellite cannot reliably measure it, which means it must be pretty far away. +Still, we can estimate its distance, based on a striking relationship between the brightness and the color of a star. A `Hertzsprung-Russel diagram `__ illustrates this. Let's make one! -In our data, column 3 contains the measurements of the apparent brightness of each star according to Gaia's sensors. -For the Hertzsprung-Russel diagram, we need to estimate the intrinsic brightness (the luminosity) of the stars, compensating for the fact that the further a star is, the dimmer it appears to us. -We will use the formula given in the `official Gaia paper `__ for this:: +Recall that in our ``data`` array, the column with index ``3`` contains the measurements of the apparent brightness of each star according to Gaia's sensors (``magnitude_g``). +For the Hertzsprung-Russel diagram, we need to estimate their true magnitude (luminosity), compensating for the fact that the further a star is, the dimmer it appears to us. +We will use the formula given in an `official Gaia paper `__ for this:: magnitude_g = data[:, 3] # apparent magnitude magnitude = magnitude_g + 5 * np.log10(parallax) - 10 # true magnitude +The :data:`numpy.log10` function is part of NumPy's vast collection of `ufuncs `__, which are functions that operate on each element of an array. + In our diagram, we want to show the color of a star on a scale from "very blue" to "very red", with yellow stars in between. We can compute this by taking the difference between the blue and red components of its color:: @@ -302,7 +303,7 @@ Then, we can compare that true brightness with its apparent brightness to deduce Creating arrays --------------- -The first step is to extract a bunch of ``(color, magnitude)`` pairs along the main sequence, free from all the outliers. +The first step is to extract a bunch of ``(color, magnitude)`` pairs along the main sequence, free from outliers. The strategy is to divide the color spectrum into 100 bins and take the median magnitude in each bin. With the vast majority of the stars being main sequence stars, the median should be very representative of the general curve of the main sequence. @@ -321,7 +322,7 @@ We could assign stars to their respective color bins using boolean masking, but bin_assignment = np.digitize(color, bins) print(bin_assignment) -The function gives for each element of the ``color`` array (as an integer index), the index of the bin it belongs to. +The function gives for each element of the ``color`` array, the index of the bin it belongs to. Since each element of the ``color`` array is the color of a star, we have effectively assigned each star to a bin. Now, we can compute for each bin, the median brightness and color of all the stars in that bin. We will do this using a ``for`` loop and use a typical pattern for creating NumPy arrays: we first collect the values into a Python list and then convert that list into a NumPy array:: @@ -342,7 +343,7 @@ We will do this using a ``for`` loop and use a typical pattern for creating NumP bin_magnitudes = np.array(bin_magnitudes) # The white dwarf stars with a color < 0.1 are all lumped in the first bin, - # so that bin is unreliable. + # so that bin is unreliable, chop it off. bin_colors = bin_colors[1:] bin_magnitudes = bin_magnitudes[1:] @@ -379,7 +380,7 @@ A typical way to do this is a machine learning technique called `ordinary least \beta = (X^T X)^{-1} X^T Y The resulting ``beta`` vector contains the optimal linear combination (in the least-squares sense) of the columns of ``X`` to predict the columns of ``Y``. -So, we create a 2D array ``X`` where the columns are ``bin_colors`` raised to different powers and set ``Y`` to ``bin_magnitude``. +So, if we create a 2D array ``X`` where the columns are ``bin_colors`` raised to different powers, and set ``Y`` to ``bin_magnitude``, we should obtain the beta values we need. The code below does this, while demonstrating how to create a 2D array by gluing 1D arrays together, and how to generate an array containing only ``1``'s:: X = np.column_stack(( @@ -403,6 +404,9 @@ Using NumPy, the Python code can look very much like the math formula:: beta = np.linalg.inv(X.T @ X) @ X.T @ Y print(beta) +.. seealso:: + `Numpy linear algebra functionality `__. + Creating and documenting functions in the scientific computing style -------------------------------------------------------------------- @@ -419,12 +423,12 @@ NumPy comes with a style guide for writing docstrings for functions called `NumP color : float | array of float, shape (n_stars,) The color of the star(s), computed as magnitude_blue - magnitude_red. beta: array of float, shape (6,) - The regression weights. + The coefficients describing the main-sequence curve. Returns ------- magnitude : float - The estimated intrinsic magnitude of the star(s). + The estimated true magnitude of the star(s). """ return ( beta[5] * color ** 5 + @@ -441,52 +445,63 @@ Exercises 2 .. challenge:: Exercises: Numpy-2 - To do the final estimation of a star's distance, we can compare our estimate of the intrinsic magnitude of a star based on its color, with the apparent magnitude of the star on our sensors. - The further away the star is, the dimmer it appears to us. - Recall this formula we used earlier:: + In this exercise, you will estimate the distance to the Large Magallanic Cloud. + + We have shown you how to estimate the true magnitude of a star based on its color and we wrote the ``predict_magnitude`` function to do it. + To predict distance, you can compare a star's estimated true magnitude with its apparent magnitude on Gaia's sensors. + The further away the star is, the dimmer it appears to Gaia. + Recall the formula we used earlier:: magnitude = magnitude_g + 5 * np.log10(parallax) - 10 - We can solve for ``parallax``, which gives us:: + When we solve for ``parallax``, we get:: estimated_parallax = np.pow(10, (magnitude - magnitude_g + 10) / 5) - 1. Write a function to predict the distance of a star given its color and apparent magnitude: + 1. Write a function to predict the distance of a star given its apparent magnitude, its color, and the ``beta`` coefficients we computed: - 1. Use the ``predict_magnitude`` function to estimate the intrinsic magnitude. - 2. Use the formula above to convert the difference between the apparent magnitude (``magnitude_g``) and intrinsic magnitude to a parallax value. + 1. Use the ``predict_magnitude`` function to estimate the true magnitude. + 2. Use the formula above to convert the difference between the apparent magnitude (``magnitude_g``) and true magnitude (``magnitude``) to a parallax value. 3. Convert the parallax value to a distance in light-years. See the lesson material above on how to do this if you don't remember. 2. Some main sequence stars from the Large Magallanic Cloud can be found in :download:`../resources/data/numpy/lmc_stars.csv`. + To read it, use :func:`numpy.genfromtext` in the same manner as we did with the ``data`` array. Predict their distance using the function you just created and take the average (:func:`numpy.mean`) as a representative value for roughly how far away the cloud is. Given that the `Milky Way `__ galaxy is about 87400 light-years across, what does the distance to the Large Magallanic Cloud tell us? .. solution:: Solution Numpy-2 - .. code:: - - def predict_distance(magnitude_g, color, beta): - """Predict a main sequence star's distance given its apparent magnitude and color. - - Parameters - ---------- - magnitude_g : float | array of float, shape (n_stars,) - The apparent magnitude of the star(s). - color : float | array of float, shape (n_stars,) - The color of the star(s), computed as magnitude_blue - magnitude_red. - beta: array of float, shape (6,) - The regression weights. - - Returns - ------- - distance : float - The estimated distance in light-years. - """ - magnitude = predict_magnitude(color, beta) - parallax = np.pow(10, (magnitude - magnitude_g + 10) / 5) - distance_parsecs = 1 / (parallax / 1000) - return distance_parsecs * 3.26156 # convert to light-years - - lmc = np.genfromtxt("../resources/data/numpy/lmc_stars.csv", delimiter=",", skip_header=1) - distance = predict_distance(lmc[:, 3], lmc[:, 4] - lmc[:, 5], beta) - print(distance.mean()) + #. .. code:: + + def predict_distance(color, magnitude_g, beta): + """Predict a main sequence star's distance given its apparent magnitude and color. + + Parameters + ---------- + magnitude_g : float | array of float, shape (n_stars,) + The apparent magnitude of the star(s). + color : float | array of float, shape (n_stars,) + The color of the star(s), computed as magnitude_blue - magnitude_red. + beta: array of float, shape (6,) + The coefficients describing the main-sequence curve. + + Returns + ------- + distance : float + The estimated distance in light-years. + """ + magnitude = predict_magnitude(color, beta) + parallax = np.pow(10, (magnitude - magnitude_g + 10) / 5) + distance_parsecs = 1 / (parallax / 1000) + return distance_parsecs * 3.26156 # convert to light-years + + #. .. code :: + + lmc = np.genfromtxt( + "../resources/data/numpy/lmc_stars.csv", delimiter=",", skip_header=1 + ) + distance = predict_distance(lmc[:, 3], lmc[:, 4] - lmc[:, 5], beta) + print(distance.mean()) + + The Large Magallanic Cloud is roughly 158000 light-years away. + This means it does not lie in our galaxy, but must be a galaxy of its own. 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