From d387e9bd89ecfb83fe3c3f8c5e5cb1914a24726f Mon Sep 17 00:00:00 2001 From: Jammy2211 Date: Sat, 11 Jul 2026 16:19:44 +0100 Subject: [PATCH 1/2] Add markdown_examples.yaml + README browse link (HowToLens chapter 1) Batch 2b of the executed-markdown rollout: registers chapter_1_introduction tutorials for generate_markdown.py and links the (to-be-generated) markdown/ index from the README. Co-Authored-By: Claude Opus 4.8 Claude-Session: https://claude.ai/code/session_01NyqcbXuZvGVTHHEWhoEWkC --- README.md | 1 + config/build/markdown_examples.yaml | 21 +++++++++++++++++++++ 2 files changed, 22 insertions(+) create mode 100644 config/build/markdown_examples.yaml diff --git a/README.md b/README.md index 11a233f..24b6c57 100644 --- a/README.md +++ b/README.md @@ -2,6 +2,7 @@ [Installation Guide](https://pyautolens.readthedocs.io/en/latest/installation/overview.html) | [PyAutoLens readthedocs](https://pyautolens.readthedocs.io/en/latest/index.html) | +[Browse Chapter 1 With Images](markdown/README.md) | [autolens_workspace](https://github.com/PyAutoLabs/autolens_workspace) diff --git a/config/build/markdown_examples.yaml b/config/build/markdown_examples.yaml new file mode 100644 index 0000000..c34a517 --- /dev/null +++ b/config/build/markdown_examples.yaml @@ -0,0 +1,21 @@ +# Curated examples rendered to executed markdown pages (markdown/) with real +# output images, for GitHub browsing. Built by PyAutoBuild's generate_markdown.py. +# Batch 2b: chapter_1_introduction (no non-linear searches — fast). +- script: scripts/chapter_1_introduction/tutorial_0_visualization.py + max_minutes: 30 +- script: scripts/chapter_1_introduction/tutorial_1_grids_and_galaxies.py + max_minutes: 30 +- script: scripts/chapter_1_introduction/tutorial_2_ray_tracing.py + max_minutes: 30 +- script: scripts/chapter_1_introduction/tutorial_3_more_ray_tracing.py + max_minutes: 30 +- script: scripts/chapter_1_introduction/tutorial_4_point_sources.py + max_minutes: 30 +- script: scripts/chapter_1_introduction/tutorial_5_lensing_formalism.py + max_minutes: 30 +- script: scripts/chapter_1_introduction/tutorial_6_data.py + max_minutes: 30 +- script: scripts/chapter_1_introduction/tutorial_7_fitting.py + max_minutes: 30 +- script: scripts/chapter_1_introduction/tutorial_8_summary.py + max_minutes: 30 From a13ce4869fc280b3deed10d77d07851222722a1b Mon Sep 17 00:00:00 2001 From: Jammy2211 Date: Sat, 11 Jul 2026 16:31:24 +0100 Subject: [PATCH 2/2] Add executed markdown pages for HowToLens Chapter 1 (browse with images) The full chapter_1_introduction (9 tutorials) executed with output images, index linked from the README. Batch 2b (HowToLens#24). Also adds the standard `setup_notebook()` line to tutorial_7_fitting (it was missing it, so it failed to execute under nbconvert). Notebook regenerated. 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a/markdown/README.md b/markdown/README.md new file mode 100644 index 0000000..2580590 --- /dev/null +++ b/markdown/README.md @@ -0,0 +1,15 @@ +# HowToLens examples, executed — browse with output images + +Every page below is the corresponding example script **fully executed**, rendered to markdown with its real output images, so you can read the examples on GitHub exactly as they run. Each page links back to the `.py` script and Jupyter notebook it was generated from. + +- [Tutorial 0: Visualization](chapter_1_introduction/tutorial_0_visualization.md) — from `scripts/chapter_1_introduction/tutorial_0_visualization.py` +- [HowToLens: Introduction](chapter_1_introduction/tutorial_1_grids_and_galaxies.md) — from `scripts/chapter_1_introduction/tutorial_1_grids_and_galaxies.py` +- [Tutorial 2: Ray Tracing](chapter_1_introduction/tutorial_2_ray_tracing.md) — from `scripts/chapter_1_introduction/tutorial_2_ray_tracing.py` +- [Tutorial 5: More Ray Tracing](chapter_1_introduction/tutorial_3_more_ray_tracing.md) — from `scripts/chapter_1_introduction/tutorial_3_more_ray_tracing.py` +- [Tutorial 4: Point Sources](chapter_1_introduction/tutorial_4_point_sources.md) — from `scripts/chapter_1_introduction/tutorial_4_point_sources.py` +- [Tutorial 5: Lensing Formalism](chapter_1_introduction/tutorial_5_lensing_formalism.md) — from `scripts/chapter_1_introduction/tutorial_5_lensing_formalism.py` +- [Tutorial 6: Data](chapter_1_introduction/tutorial_6_data.md) — from `scripts/chapter_1_introduction/tutorial_6_data.py` +- [Tutorial 7: Fitting](chapter_1_introduction/tutorial_7_fitting.md) — from `scripts/chapter_1_introduction/tutorial_7_fitting.py` +- [Tutorial 9: Summary](chapter_1_introduction/tutorial_8_summary.md) — from `scripts/chapter_1_introduction/tutorial_8_summary.py` + +These pages are regenerated manually by PyAutoBuild's `generate_markdown.py` when a curated script changes. diff --git a/markdown/chapter_1_introduction/tutorial_0_visualization.md b/markdown/chapter_1_introduction/tutorial_0_visualization.md new file mode 100644 index 0000000..3b46d13 --- /dev/null +++ b/markdown/chapter_1_introduction/tutorial_0_visualization.md @@ -0,0 +1,193 @@ +> ✏️ **This page is auto-generated from [`scripts/chapter_1_introduction/tutorial_0_visualization.py`](../../scripts/chapter_1_introduction/tutorial_0_visualization.py) — do not edit it directly.** +> It shows the example fully executed, with its real output images. +> Run it yourself via the [Python script](../../scripts/chapter_1_introduction/tutorial_0_visualization.py) or the [Jupyter notebook](../../notebooks/chapter_1_introduction/tutorial_0_visualization.ipynb). + +Tutorial 0: Visualization +========================= + +In this tutorial, we quickly cover visualization in **PyAutoLens** and make sure images display +clearly in your Jupyter notebook and on your computer screen. + +__Contents__ + +- **Directories:** **PyAutoLens assumes** the working directory is `autolens_workspace` on your hard-disk. +- **Dataset:** Load and plot the strong lens dataset. +- **Subplots:** In addition to plotting individual figures, **PyAutoLens** can plot `subplots` which show multiple. +- **Plot Customization:** Does the figure display correctly on your computer screen? +- **Overlays:** Overlays such as critical curves and image positions are added using the `lines=` and `positions=`. +- **Wrap Up:** Summary of the script and next steps. + + +```python + +from autoconf import jax_wrapper # Sets JAX environment before other imports + +from autoconf import setup_notebook; setup_notebook() +``` + + Working Directory has been set to `HowToLens` + + +If the printed working directory does not match the workspace path on your computer, you can manually set it +as follows (the example below shows the path I would use on my laptop. The code is commented out so you do not +use this path in this tutorial! + + +```python +# workspace_path = "/Users/Jammy/Code/PyAuto/autolens_workspace" +# #%cd $workspace_path +# print(f"Working Directory has been set to `{workspace_path}`") +``` + +__Dataset__ + +The `dataset_path` specifies where the dataset is located, which is the +directory `autolens_workspace/dataset/imaging/simple__no_lens_light`. + +There are many example simulated images of strong lenses in this directory that will be used throughout the +**HowToLens** lectures. + + +```python +from pathlib import Path + +import autolens as al +import autolens.plot as aplt + +dataset_path = Path("dataset") / "imaging" / "simple__no_lens_light" +``` + +We now load this dataset from .fits files and create an instance of an `Imaging` object. + + +```python +dataset = al.Imaging.from_fits( + data_path=dataset_path / "data.fits", + noise_map_path=dataset_path / "noise_map.fits", + psf_path=dataset_path / "psf.fits", + pixel_scales=0.1, +) +``` + +We can plot an image with `aplt.plot_array()`, passing the data array and a title. + + +```python +aplt.plot_array(array=dataset.data, title="Dataset Image") +``` + + + +![png](tutorial_0_visualization_files/tutorial_0_visualization_9_0.png) + + + +__Subplots__ + +In addition to plotting individual figures, **PyAutoLens** can plot `subplots` which show multiple +views of the dataset at once. + +The `aplt.subplot_imaging_dataset()` function plots the data, noise-map and PSF together. + + +```python +aplt.subplot_imaging_dataset(dataset=dataset) +``` + + + +![png](tutorial_0_visualization_files/tutorial_0_visualization_11_0.png) + + + +__Plot Customization__ + +Does the figure display correctly on your computer screen? + +If not, the default matplotlib settings can be customized via the config files in: + + autolens_workspace/config/visualize/ + +Key config entries: + + - `mat_wrap.yaml` -> Figure -> figure: -> figsize + - `mat_wrap.yaml` -> YLabel -> figure: -> fontsize + - `mat_wrap.yaml` -> XLabel -> figure: -> fontsize + - `mat_wrap.yaml` -> TickParams -> figure: -> labelsize + - `mat_wrap.yaml` -> YTicks -> figure: -> labelsize + - `mat_wrap.yaml` -> XTicks -> figure: -> labelsize + +For quick one-off adjustments you can pass `title=`, `colormap=`, and `use_log10=` directly: + + +```python +aplt.plot_array(array=dataset.data, title="Dataset Image (Log10)", use_log10=True) +``` + + + +![png](tutorial_0_visualization_files/tutorial_0_visualization_13_0.png) + + + +__Overlays__ + +Overlays such as critical curves and image positions are added using the `lines=` and `positions=` +keyword arguments. + +For example, we can compute the critical curves of a tracer and overlay them on the image. + + +```python +grid = al.Grid2D.uniform(shape_native=(100, 100), pixel_scales=0.05) + +lens_galaxy = al.Galaxy( + redshift=0.5, + mass=al.mp.Isothermal(centre=(0.0, 0.0), einstein_radius=1.6, ell_comps=(0.0, 0.0)), +) + +source_galaxy = al.Galaxy( + redshift=1.0, + bulge=al.lp.SersicCoreSph( + centre=(0.0, 0.0), intensity=1.0, effective_radius=0.5, sersic_index=2.0 + ), +) + +tracer = al.Tracer(galaxies=[lens_galaxy, source_galaxy]) + +tangential_critical_curve_list = al.LensCalc.from_tracer( + tracer=tracer +).tangential_critical_curve_list_from(grid=grid) + +aplt.plot_array( + array=tracer.image_2d_from(grid=grid), + title="Tracer Image with Critical Curves", + lines=tangential_critical_curve_list, +) +``` + + + +![png](tutorial_0_visualization_files/tutorial_0_visualization_15_0.png) + + + +__Wrap Up__ + +Throughout the lectures you'll see lots more visuals plotted on figures and subplots. + +The key plotting functions you'll use are: + + - `aplt.plot_array(array, title, ...)` — plot any 2D array. + - `aplt.plot_grid(grid, title, ...)` — plot a 2D grid of coordinates. + - `aplt.subplot_imaging_dataset(dataset)` — multi-panel dataset overview. + - `aplt.subplot_tracer(tracer, grid)` — multi-panel tracer overview. + - `aplt.subplot_fit_imaging(fit)` — multi-panel fit overview. + +Great! Hopefully, visualization in **PyAutoLens** is displaying nicely for us to get on with the +**HowToLens** lecture series. + + +```python + +``` diff --git a/markdown/chapter_1_introduction/tutorial_0_visualization_files/tutorial_0_visualization_11_0.png b/markdown/chapter_1_introduction/tutorial_0_visualization_files/tutorial_0_visualization_11_0.png new file mode 100644 index 0000000000000000000000000000000000000000..158af60a1f062e180e2cc90e7a7d159bd2a28429 GIT binary patch literal 756499 zcmc$`WmHvb_%8}b3y5?{7=UzlNMq0-NQZ)ScS=iIhys!-2uMhGNtZ~2ln9G%q~Si( z{Xgf9anAjIKQP=I)|zY1H=gHL?|QDTsziuSgO7%WMtE0QUK0%s`y75yxY+QUGBz?_ z_(#l5LC;Og(bCP+%;gE1ikX{}ouiwbwK=256Bk!&M+bf`ULGz{4#uZ$ZceUZ+}!s6 z`xRV{E>_&kii3~fDtJ!Hk6h8vu9zV|=sD8a)@bNxXm{mhw7pW+K6s{(Dg140m?HJG zawW3Ec$<=ikNZl#2me@{Al;nEs+vFAs&Dy?_zguKcD6@%Zn8SdxJ(!cp;DOL&(5hl zO+9@QnY>s>4tB?{$f@bvPXDYn? zH{3fcax1h-?WnZCJ~inxc{eu!;c2h==M<=SIk~y!ov&_5Y=0+B2|Pc#dGn@Md(!~N z@z+Ns;oaN~DR178TX;k%t*pUvmurOsv(~nFC50Af_?D_K7kGh%uO%G+o 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z^OJ9>T27MCbu%XI4w8A3>R{Eo|IPF#ctK|0GV=fO%ca`P8q3?;lvOWK@MnvSgZ0gg H-ZB3JFXLyW literal 0 HcmV?d00001 diff --git a/markdown/chapter_1_introduction/tutorial_1_grids_and_galaxies.md b/markdown/chapter_1_introduction/tutorial_1_grids_and_galaxies.md new file mode 100644 index 0000000..9b3a8ba --- /dev/null +++ b/markdown/chapter_1_introduction/tutorial_1_grids_and_galaxies.md @@ -0,0 +1,575 @@ +> ✏️ **This page is auto-generated from [`scripts/chapter_1_introduction/tutorial_1_grids_and_galaxies.py`](../../scripts/chapter_1_introduction/tutorial_1_grids_and_galaxies.py) — do not edit it directly.** +> It shows the example fully executed, with its real output images. +> Run it yourself via the [Python script](../../scripts/chapter_1_introduction/tutorial_1_grids_and_galaxies.py) or the [Jupyter notebook](../../notebooks/chapter_1_introduction/tutorial_1_grids_and_galaxies.ipynb). + +HowToLens: Introduction +======================= + +A strong gravitational lens is a system where two (or more) galaxies align perfectly down our line of sight from Earth +such that the foreground galaxy's mass curves space-time in on itself, such that the light of a background source galaxy +is deflected and magnified. This means we can see the background source galaxy multiple times, as multiple arcs or rings, +because multiple paths through the foreground galaxy's mass are taken by the source's light. + +Here is a schematic of a strong gravitational lens: + +![Schematic of Gravitational Lensing](https://raw.githubusercontent.com/PyAutoLabs/PyAutoLens/main/docs/overview/images/overview_1_lensing/schematic.jpg) +**Credit: F. Courbin, S. G. Djorgovski, G. Meylan, et al., Caltech / EPFL / WMKO** +https://www.astro.caltech.edu/~george/qsolens/ + +Today, Astronomers use computer software, statistical algorithms and image processing techniques analyse the light of +these strong gravitational lenses. They are used for a wide range of scientific studies, including studying dark matter +in the foreground lens galaxies, the morphology and structure of the background source galaxies, and even the expansion +of the Universe itself. + +The **HowToLens** series of tutorials will teach you how to perform this analysis yourself, using the open-source +software package **PyAutoLens**. By the end of the **HowToLens** series, you'll be able to take an image of +strong lens and study it using the same techniques that professional astronomers use today. + +Tutorial 1: Grids And Galaxies +============================== + +In this tutorial, we will introduce the first fundamental concepts and quantities used to study strong lenses. +These concepts will enable us to create images of galaxies and analyze how their light is distributed across space. +Additionally, we will explore how adjusting various properties of galaxies can alter their appearance. For instance, +we can change the size of a galaxy, rotate it, or modify its brightness. + +To create these images, we first need to define 2D grids of \((y, x)\) coordinates. We will shift and rotate these +grids to manipulate the appearance of the galaxy in the generated images. The grid will serve as the input for light +profiles, which are analytic functions that describe the distribution of a galaxy's light. By evaluating these light +profiles on the grid, we can effectively generate images that represent the structure and characteristics of galaxies. + +This tutorial won't yet perform any lensing calculations, which are introduced in the next tutorial. + +Here is an overview of what we'll cover in this tutorial: + +- **Grids**: We'll create a uniform grid of $(y,x)$ coordinates and show how it can be used to measure the light of a galaxy. +- **Geometry**: We'll show how to shift and rotate a grid, and convert it to elliptical coordinates. +- **Light Profiles**: We'll introduce light profiles, analytic functions that describe how a galaxy's light is distributed. +- **Galaxies**: We'll create galaxies containing light profiles and show how to compute their image. +- **Units**: We'll show how to convert the units of a galaxy's image to physical units like kiloparsecs. + + +The imports below are required to run the HowToLens tutorials in a Jupiter notebook. They also import the +`autolens` package and the `autolens.plot` module which are used throughout the tutorials. + +__Contents__ + +- **Grids:** A `Grid2D` is a set of two-dimensional $(y,x)$ coordinates that represent points in space where we. +- **Geometry:** The above grid is centered on the origin (0.0", 0.0"). +- **Light Profiles:** Galaxies are collections of stars, gas, dust, and other astronomical objects that emit light. +- **One Dimension Projection:** We often want to calculative 1D quantities of a light profile, for example to plot how its light. + + +```python + +from autoconf import jax_wrapper # Sets JAX environment before other imports + +from autoconf import setup_notebook; setup_notebook() + +import matplotlib.pyplot as plt +import numpy as np + +import autoarray as aa +import autolens as al +import autolens.plot as aplt +``` + + Working Directory has been set to `HowToLens` + + +__Grids__ + +A `Grid2D` is a set of two-dimensional $(y,x)$ coordinates that represent points in space where we evaluate the +light emitted by a galaxy. + +Each coordinate on the grid is referred to as a 'pixel'. This is because we use the grid to measure the brightness of a +galaxy at each of these coordinates, allowing us to create an image of the galaxy. + +Grids are defined in units of 'arc-seconds' ("). An arc-second is a unit of angular measurement used by astronomers to +describe the apparent size of objects in the sky. + +The `pixel_scales` parameter sets how many arc-seconds each pixel represents. For example, if `pixel_scales=0.1`, +then each pixel covers 0.1" of the sky. + +We create a uniform 2D grid of 101 x 101 pixels with a pixel scale of 0.1", corresponding to an area +of 10.1" x 10.1", spanning from -5.05" to 5.05" in both the y and x directions. + + +```python +grid = al.Grid2D.uniform( + shape_native=( + 101, + 101, + ), # The dimensions of the grid, which here is 100 x 100 pixels. + pixel_scales=0.1, # The conversion factor between pixel units and arc-seconds. +) +``` + +We can visualize this grid as a uniform grid of dots, each representing a coordinate where the light is measured. + + +```python +aplt.plot_grid(grid=grid, title="Uniform Grid of Coordinates") +``` + + + +![png](tutorial_1_grids_and_galaxies_files/tutorial_1_grids_and_galaxies_5_0.png) + + + +Each coordinate in the grid corresponds to an arc-second position. Below, we print a few of these coordinates to see +the values. + + +```python +print("(y,x) pixel 0:") +print(grid.native[0, 0]) # The coordinate of the first pixel. +print("(y,x) pixel 1:") +print(grid.native[0, 1]) # The coordinate of the second pixel. +print("(y,x) pixel 2:") +print(grid.native[0, 2]) # The coordinate of the third pixel. +print("(y,x) pixel 100:") +print(grid.native[1, 0]) # The coordinate of the 100th pixel. +print("...") +``` + + (y,x) pixel 0: + [ 5. -5.] + (y,x) pixel 1: + [ 5. -4.9] + (y,x) pixel 2: + [ 5. -4.8] + (y,x) pixel 100: + [ 4.9 -5. ] + ... + + +Grids have two internal representations, `native` and `slim`: + +- `native`: A 2D array with shape [total_y_pixels, total_x_pixels, 2], where the 2 corresponds to the (y,x) coordinates. +- `slim`: A 1D array with shape [total_y_pixels * total_x_pixels, 2], where the coordinates are 'flattened' into a single list. + +These formats are useful for different calculations and plotting. Here, we show the same coordinate using both formats. + + +```python +print("(y,x) pixel 0 (accessed via native):") +print(grid.native[0, 0]) +print("(y,x) pixel 0 (accessed via slim 1D):") +print(grid.slim[0]) +``` + + (y,x) pixel 0 (accessed via native): + [ 5. -5.] + (y,x) pixel 0 (accessed via slim 1D): + [ 5. -5.] + + +We can also check the shapes of the `Grid2D` object in both `native` and `slim` formats. For this grid, +the `native` shape is (101, 101, 2) and the `slim` shape is (10201, 2). + + +```python +print(grid.native.shape) +print(grid.slim.shape) +``` + + (101, 101, 2) + (10201, 2) + + +For the HowToLens tutorials, you don't need to fully understand why grids have both native and slim representations. +Just note that both are used for calculations and plotting. + +*Exercise*: Try creating grids with different shapes and pixel scales using the `al.Grid2D.uniform()` function above. +Observe how the grid coordinates change when you adjust `shape_native` and `pixel_scales`. + +__Geometry__ + +The above grid is centered on the origin (0.0", 0.0"). Sometimes, we need to shift the grid to be centered on a +specific point, like the center of a galaxy. + +We can shift the grid to a new center, (y_c, x_c), by subtracting this center from each coordinate. + + +```python +centre = (0.3, 0.5) # Shifting the grid to be centered at y=1.0", x=2.0". + +grid_shifted = grid +grid_shifted[:, 0] = grid_shifted[:, 0] - centre[0] # Shift in y-direction. +grid_shifted[:, 1] = grid_shifted[:, 1] - centre[1] # Shift in x-direction. + +print("(y,x) pixel 0 After Shift:") +print(grid_shifted.native[0, 0]) # The coordinate of the first pixel after shifting. +``` + + (y,x) pixel 0 After Shift: + [ 4.7 -5.5] + + +The grid is now centered around (0.3", 0.5"). We can plot the shifted grid to see this change. + +*Exercise*: Try shifting the grid to a different center, for example (0.0", 0.0") or (2.0", 3.0"). Observe how the +center of the grid changes when you adjust the `centre` variable. + + +```python +aplt.plot_grid(grid=grid_shifted, title="Grid Centered Around (0.3, 0.5)") +``` + + + +![png](tutorial_1_grids_and_galaxies_files/tutorial_1_grids_and_galaxies_15_0.png) + + + +Next, we can rotate the grid by an angle `phi` (in degrees). The rotation is counter-clockwise from the positive x-axis. + +To rotate the grid: + +1. Calculate the distance `radius` of each coordinate from the origin using $r = \sqrt{y^2 + x^2}$. +2. Determine the angle `theta` counter clockwise from the positive x-axis using $\theta = \arctan(y / x)$. +3. Adjust `theta` by the rotation angle and convert back to Cartesian coordinates via $y = r \sin(\theta)$ and $x = r \cos(\theta)$. + + +```python +angle_degrees = 60.0 + +y = grid_shifted[:, 0] +x = grid_shifted[:, 1] + +radius = np.sqrt(y**2 + x**2) +theta = np.arctan2(y, x) - np.radians(angle_degrees) + +grid_rotated = grid_shifted +grid_rotated[:, 0] = radius * np.sin(theta) +grid_rotated[:, 1] = radius * np.cos(theta) + +print("(y,x) pixel 0 After Rotation:") +print(grid_rotated.native[0, 0]) # The coordinate of the first pixel after rotation. +``` + + (y,x) pixel 0 After Rotation: + [7.11313972 1.3203194 ] + + +The grid has now been rotated 60 degrees counter-clockwise. We can plot it to see the change. + +*Exercise*: Try rotating the grid by a different angle, for example 30 degrees or 90 degrees. Observe how the grid +changes when you adjust the `angle_degrees` variable. + + +```python +aplt.plot_grid(grid=grid_rotated, title="Grid Rotated 60 Degrees") +``` + + + +![png](tutorial_1_grids_and_galaxies_files/tutorial_1_grids_and_galaxies_19_0.png) + + + +Next, we convert the rotated grid to elliptical coordinates using: + +$\eta = \sqrt{(x_r)^2 + (y_r)^2/q^2}$ + +Where `q` is the axis-ratio of the ellipse and `(x_r, y_r)` are the rotated coordinates. + +Elliptical coordinates are a system used to describe positions in relation to an ellipse rather than a circle. They +are particularly useful in astronomy when dealing with objects like galaxies, which often have elliptical shapes +due to their inclination or intrinsic shape. + +*Exercise*: Try converting the grid to elliptical coordinates using a different axis-ratio, for example 0.3 or 0.8. +What happens to the grid when you adjust the `axis_ratio` variable? + + +```python +axis_ratio = 0.5 +eta = np.sqrt((grid_rotated[:, 0]) ** 2 + (grid_rotated[:, 1]) ** 2 / axis_ratio**2) +``` + +Above, the angle $\phi$ (in degrees) was used to rotate the grid, and the axis-ratio $q$ was used to convert the grid +to elliptical coordinates. + +From now on, we'll describe ellipticity using "elliptical components" $\epsilon_{1}$ and $\epsilon_{2}$, calculated +from $\phi$ and $q$: + +$\epsilon_{1} = \frac{1 - q}{1 + q} \sin(2\phi)$ +$\epsilon_{2} = \frac{1 - q}{1 + q} \cos(2\phi)$ + +We'll refer to these as `ell_comps` in the code for brevity. + +Future tutorials will explain why $\epsilon_{1}$ and $\epsilon_{2}$ are preferred over $q$ and $\phi$. + +*Exercise*: Try computing the elliptical components from the axis-ratio and angle above. What happens to the elliptical +components when you adjust the `axis_ratio` and `angle_degrees` variables? + + +```python +fac = (1 - axis_ratio) / (1 + axis_ratio) +epsilon_y = fac * np.sin(2 * np.radians(angle_degrees)) +epsilon_x = fac * np.cos(2 * np.radians(angle_degrees)) + +ell_comps = (epsilon_y, epsilon_x) + +print("Elliptical Components:") +print(ell_comps) + +``` + + Elliptical Components: + (np.float64(0.28867513459481287), np.float64(-0.16666666666666657)) + + +__Light Profiles__ + +Galaxies are collections of stars, gas, dust, and other astronomical objects that emit light. Astronomers study this +light to understand various properties of galaxies. + +To model the light of a galaxy, we use light profiles, which are mathematical functions that describe how a galaxy's +light is distributed across space. By applying these light profiles to 2D grids of $(y, x)$ coordinates, we can +create images that represent a galaxy's luminous emission. + +A commonly used light profile is the `Sersic` profile, which is widely adopted in astronomy for representing galaxy +light. The `Sersic` profile is defined by the equation: + +$I_{\rm Ser} (\eta_{\rm l}) = I \exp \left\{ -k \left[ \left( \frac{\eta}{R} \right)^{\frac{1}{n}} - 1 \right] \right\}$ + +In this equation: + + - $\eta$ represents the elliptical coordinates of the profile in arc-seconds (refer to earlier sections for elliptical coordinates). + - $I$ is the intensity normalization of the profile, given in arbitrary units, which controls the overall brightness of the Sersic profile. + - $R$ is the effective radius in arc-seconds, which determines the size of the profile. + - $n$ is the Sersic index, which defines how 'steep' the profile is, influencing the concentration of light. + - $k$ is a constant that ensures half the light of the profile lies within the radius $R$, where $k = 2n - \frac{1}{3}$. + +We can evaluate this function using values for $(\eta, I, R, n)$ to calculate the intensity of the profile at +a particular elliptical coordinate. + + +```python +elliptical_coordinate = ( + 0.5 # The elliptical coordinate where we compute the intensity, in arc-seconds. +) +intensity = 1.0 # Intensity normalization of the profile in arbitrary units. +effective_radius = 2.0 # Effective radius of the profile in arc-seconds. +sersic_index = 1.0 # Sersic index of the profile. +k = 2 * sersic_index - ( + 1.0 / 3.0 +) # Calculating the constant k, note that this is an approximation. + +# Calculate the intensity of the Sersic profile at a specific elliptical coordinate. +sersic_value = np.exp( + -k * ((elliptical_coordinate / effective_radius) ** (1.0 / sersic_index) - 1.0) +) + +print("Intensity of Sersic Light Profile at Elliptical Coordinate 0.5:") +print(sersic_value) +``` + + Intensity of Sersic Light Profile at Elliptical Coordinate 0.5: + 3.4903429574618414 + + +The calculation above gives the intensity of the Sersic profile at an elliptical coordinate of 0.5. + +To create a complete image of the Sersic profile, we can evaluate the intensity at every point in our grid of +elliptical coordinates. + + +```python +sersic_image = np.exp(-k * ((eta / effective_radius) ** (1.0 / sersic_index) - 1.0)) +``` + +When we plot the resulting image, we can see how the properties of the grid affect its appearance: + + - The peak intensity is at the position (0.3", 0.5"), where we shifted the grid. + - The image is elongated along a 60° counter-clockwise angle, corresponding to the rotation of the grid. + - The image has an elliptical shape, consistent with the axis ratio of 0.5. + +This demonstrates how the geometry of the grid directly influences the appearance of the light profile. + +*Exercise*: Try changing the values of `centre`, `ell_comps`, `effective_radius`, and `sersic_index` above. +Observe how these adjustments change the Sersic profile image. + + +```python +aplt.plot_array( + array=aa.Array2D(values=sersic_image, mask=grid.mask), title="Sersic Image" +) +``` + + + +![png](tutorial_1_grids_and_galaxies_files/tutorial_1_grids_and_galaxies_29_0.png) + + + +Instead of manually handling these transformations, we can use `LightProfile` objects from the `light_profile` +module (`lp`) for faster and more efficient calculations. + +Below, we define a `Sersic` light profile using the `Sersic` object. We can print the profile to display its parameters. + + +```python +sersic_light_profile = al.lp.Sersic( + centre=(0.0, 0.0), + ell_comps=(0.0, 0.1), + intensity=1.0, + effective_radius=2.0, + sersic_index=1.0, +) + +print(sersic_light_profile) +``` + + Sersic + centre: (0.0, 0.0) + ell_comps: (0.0, 0.1) + intensity: 1.0 + effective_radius: 2.0 + sersic_index: 1.0 + + +With this `Sersic` light profile, we can create an image by passing a grid to its `image_2d_from` method. + +The calculation will internally handle all the coordinate transformations and intensity evaluations we performed +manually earlier, making it much simpler. + +The `Sersic` profile we created just above is different from the one we used to manually compute the image, +so the image will look different. However, the process is the same. + + +```python +image = sersic_light_profile.image_2d_from(grid=grid) + +aplt.plot_array(array=image, title="Sersic Image via Light Profile") +``` + + + +![png](tutorial_1_grids_and_galaxies_files/tutorial_1_grids_and_galaxies_33_0.png) + + + +The `image` is returned as an `Array2D` object. Similar to a `Grid2D`, it has two forms: + + - `native`: A 2D array with shape [total_y_image_pixels, total_x_image_pixels]. + - `slim`: A 1D array that flattens this data into shape [total_y_image_pixels * total_x_image_pixels]. + +The `native` form is often used for visualizations, while the `slim` form can be useful for certain calculations. + + +```python +print("Intensity of pixel 0:") +print(image.native[0, 0]) +print("Intensity of pixel 1:") +print(image.slim[1]) +``` + + Intensity of pixel 0: + 0.01336649114209766 + Intensity of pixel 1: + 0.014020623587576737 + + +To visualize the light profile's image, we use `aplt.plot_array`. + +We provide it with the light profile and the grid, which are used to create and plot the image. + + +```python +aplt.plot_array( + array=sersic_light_profile.image_2d_from(grid=grid), title="Image via Light Profile" +) +``` + + + +![png](tutorial_1_grids_and_galaxies_files/tutorial_1_grids_and_galaxies_37_0.png) + + + +__One Dimension Projection__ + +We often want to calculative 1D quantities of a light profile, for example to plot how its light changes as +a function of radius. + +To do this, we must still input a 2D grid into the `image_2d_from` method, therefore we create a project 2D +radial grid as follows which has shape [Number_of_1d_coordinates, 2] and where all [:,0] entries are the same. + +A simple example of such a grid is as follows with 4 1D coordinates is: + + +```python +grid_2d_projected = al.Grid2DIrregular( + [ + [1.000000e-06, 1.000000e-06], + [1.000000e-06, 1.000001e00], + [1.000000e-06, 2.000001e00], + [1.000000e-06, 3.000001e00], + ] +) +``` + +As in this example, we often already have a 2D grid we are using to calculate images of a ligth profile +and it would be convenient to simply create `grid_2d_projected` from that. + +For example, we may want the project grid which traces it major axis in uniform radial steps. + +This is easily computed using the `grid_2d_radial_project_from` function and passing the `centre` and `angle` +of a light profile we can make it align with the light profile itself. + +Note how in this example the two galaxy bulges are not rotationally aligned but we aligned the projected +grid with the first galaxy. The centres are aligned, but if they were not that would cause similar +issues. + + +```python +grid_2d_projected = grid.grid_2d_radial_projected_from( + centre=sersic_light_profile.centre, angle=sersic_light_profile.angle() +) + +image_1d = sersic_light_profile.image_2d_from(grid=grid_2d_projected) +``` + +We can now plot the 1D radial profile of the light profile. This profile shows how the intensity of the light +changes as a function of distance from the profile's center. This is a more informative way to visualize the light p +rofile's distribution. + +When we plot 1D quantities, we do not use built-in plotting functions as in 2D, but instead use standard +matplotlib functionality. + +The reason is partly that 1D plotting is simple, but also because 1D plots have many different decisions +about what is plotted and how they are computed, meaning its better to give the user full control. + +**Exercise**: Try plotting the 1D radial profile of Sersic profiles with different effective radii and Sersic indices. +Does the 1D representation show more clearly how the light distribution changes with these parameters? + + +```python +plt.plot(grid_2d_projected[:, 1], image_1d) +plt.xlabel("Radius (arcseconds)") +plt.ylabel("Luminosity") +plt.show() +plt.close() + +``` + + + +![png](tutorial_1_grids_and_galaxies_files/tutorial_1_grids_and_galaxies_43_0.png) + + + +Since galaxy light distributions often cover a wide range of values, they are typically better visualized on a log10 +scale. This approach helps highlight details in the faint outskirts of a light profile. + +The `plot_array`/`subplot_\*` object has a `use_log10` option that applies this transformation automatically. 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the [Jupyter notebook](../../notebooks/chapter_1_introduction/tutorial_2_ray_tracing.ipynb). + +Tutorial 2: Ray Tracing +======================= + +Strong gravitational lensing occurs when the mass of a foreground galaxy (or galaxies) curves space-time around it, +causing light rays from a background source to appear deflected. + +The process of ray tracing calculates how much the path of each light ray is bent by the mass of the foreground galaxy. +It then traces the paths of these light rays back to the observer, allowing us to determine how the source appears +distorted. As a result, the images of the source may appear as arcs, multiple images, or other complex patterns. + +In the previous tutorial, we introduced **light profiles**, which are analytic functions that describe the distribution +of light from a galaxy. In this tutorial, we will focus on **mass profiles**, which are analytic functions that +describe the mass distribution within a galaxy. A key quantity derived from these profiles is the **deflection angle**, +which quantifies how light is deflected by the mass of the galaxy at any point in space. + +Grids were crucial in the previous tutorial, as they enabled us to compute the light profile of a galaxy at every +coordinate point. In ray tracing, grids are equally important because they are used to calculate the deflection +angles of light rays caused by a mass profile and to map the source's light rays back to the observer. + +Let’s revisit the schematic of strong lensing: + +![Schematic of Gravitational Lensing](https://i.imgur.com/zB6tIdI.jpg) + +As observers, we do not see the true appearance of the source (e.g., a round blob of light). Instead, we only +perceive its light after it has been deflected and lensed by the foreground galaxies. This resulting image is known +as the **observed image** or **image-plane image**, which we will produce through the ray-tracing process. + +The schematic above uses the terms "image-plane" and "source-plane." In the context of gravitational lensing, a "plane" +refers to a collection of galaxies located at the same redshift, meaning they are physically aligned parallel to one +another. + +In this tutorial, we will create a strong lensing system consisting of planes similar to the one depicted above. While +a plane can contain multiple galaxies, we will focus on a simple scenario with just one lens galaxy and one source +galaxy. + +Here is an overview of what we'll cover in this tutorial: + +- **Grid**: How the 2D grids that were important for evaluating light profiles are equally important + for ray-tracing calculations. + +- **Mass Profiles**: Introduce mass profiles, which describe the mass distribution of galaxies and are used to + calculate deflection angles. + +- **Ray Tracing Grids**: How to map the light rays from the image-plane to the source-plane using the + deflection angles and **lens equation**. + +- **Ray Tracing Images**: How to evaluate the lensed image of a source galaxy after it has been + gravitationally lensed. + +- **Galaxies**: How to include both light and mass profiles in a single `Galaxy` object, and therefore +construct realistic lens and source galaxies. + +- **Tracer**: Introduce the `Tracer` object, which automates the ray-tracing process and allows us to compute + images of the entire lens system. + +- **Mappings**: Visualize how image pixels map to the source plane and vice versa using the `lines=`/`positions=` overlays object. + +__Contents__ + +- **Grid:** In the previous tutorial, we created 2D grids of (y,x) coordinates and showed how shifting and. +- **Mass Profiles:** To perform lensing calculations, we use mass profiles available in the `mass_profile` module. + + +```python + +from autoconf import jax_wrapper # Sets JAX environment before other imports + +from autoconf import setup_notebook; setup_notebook() + +import matplotlib.pyplot as plt +import autolens as al +import autoarray as aa +import autolens.plot as aplt + +``` + + Working Directory has been set to `HowToLens` + + +__Grid__ + +In the previous tutorial, we created 2D grids of (y,x) coordinates and showed how shifting and rotating these grids +is crucial for evaluating the light profiles of galaxies. + +Grids are also essential for performing ray-tracing calculations. The coordinates in the grid are deflected by the +lens galaxy, and these deflected coordinates are used in ray-tracing calculations. + +Now, let’s create the grid for this tutorial, which we’ll call the `image_plane_grid`. It represents the grid of +coordinates in the image plane, before the light is deflected by the lens galaxy. This grid is uniform, meaning +every coordinate is evenly spaced. However, this uniformity will change after ray-tracing, as the light rays are +mapped to the source plane. + + +```python +image_plane_grid = al.Grid2D.uniform(shape_native=(101, 101), pixel_scales=0.1) +``` + +__Mass Profiles__ + +To perform lensing calculations, we use mass profiles available in the `mass_profile` module (accessible via `al.mp`). + +A mass profile is an analytic function that describes the mass distribution within a galaxy. It is used to calculate +deflection angles and other quantities like surface density and gravitational potential. + +In gravitational lensing, deflection angles describe how a mass bends light by curving space-time. + +We will start with a simple mass profile, `IsothermalSph`, which represents a spherically symmetric isothermal +mass distribution. This profile has two main properties: its center (the origin of the coordinate system) and its +Einstein radius (which indicates the galaxy's mass and how much it bends light rays). + + +```python +sis_mass_profile = al.mp.IsothermalSph( + centre=(0.0, 0.0), # The (y,x) arc-second coordinates of the profile's center. + einstein_radius=1.6, # The Einstein radius of the profile in arc-seconds. +) +print(sis_mass_profile) +``` + + IsothermalSph + centre: (0.0, 0.0) + ell_comps: (0.0, 0.0) + einstein_radius: 1.6 + slope: 2.0 + core_radius: 0.0 + + +In the previous tutorial, we used the `image_2d_from` method to compute the image of a light profile by evaluating +its intensity at each (y,x) coordinate on the grid. + +Mass profiles have a similar method called `deflections_yx_2d_from`, which calculates the deflection angles at +every (y,x) coordinate on the grid in units of arc-seconds. + + +```python +deflections = sis_mass_profile.deflections_yx_2d_from(grid=image_plane_grid) +``` + +Like grids and arrays, the deflection angles can be accessed using the `native` and `slim` attributes. These are +structured similarly to a `Grid2D` object: + +- **native**: A 2D array with shape \([total_y_pixels, total_x_pixels, 2]\), where the last dimension represents the + (y,x) deflection components. + +- **slim**: A 1D array with shape \([total_y_pixels * total_x_pixels, 2]\), where the coordinates are flattened + into a single list. + + +```python +print("Deflection angles of pixel 0:") +print(deflections.native[0, 0]) +print("Deflection angles of pixel 1:") +print(deflections.slim[1]) +``` + + Deflection angles of pixel 0: + [ 1.13137085 -1.13137085] + Deflection angles of pixel 1: + [ 1.14274054 -1.11988573] + + +There is an important difference between a grid and deflection angles. A `Grid2D` is a set of coordinates, while +deflection angles are 2D vectors. This means that each deflection angle is defined at a specific (y,x) coordinate +but has two components: a y and an x value, which vary across the grid. + +This is why the method is called `deflections_yx_2d_from`—the `yx` signifies that these are 2D vectors with both +y and x components. + +The deflection angles are stored in a `VectorYX2D` data structure: + + +```python +print(type(deflections)) +``` + + + + +This structure includes a `grid`, which represents the `Grid2D` of coordinates where the deflection angles are +calculated (in this case, the `image_plane_grid` we defined earlier). It also has vector-specific methods, +such as `magnitude`, which calculates the magnitude of each deflection vector using \((x^2 + y^2)^{0.5}\). + + +```python +print("Deflection angle's `Grid2D` at pixel 0:") +print(deflections.grid.native[0, 0]) +print("Deflection angle magnitude at pixel 0:") +print(deflections.magnitudes.native[0, 0]) +``` + + Deflection angle's `Grid2D` at pixel 0: + [ 5. -5.] + Deflection angle magnitude at pixel 0: + 1.6 + + +We can use `aplt.plot_array` to visualize the deflection angles, which displays the y and x components separately. + +On this plot, you’ll see yellow and white lines called **critical curves**. These curves are important in lensing +and will be explained in detail in the next tutorial. + + +```python +deflections = sis_mass_profile.deflections_yx_2d_from(grid=image_plane_grid) +deflections_y = aa.Array2D(values=deflections.slim[:, 0], mask=image_plane_grid.mask) +aplt.plot_array(array=deflections_y, title="Deflections Y") +deflections = sis_mass_profile.deflections_yx_2d_from(grid=image_plane_grid) +deflections_x = aa.Array2D(values=deflections.slim[:, 1], mask=image_plane_grid.mask) +aplt.plot_array(array=deflections_x, title="Deflections X") +``` + + + +![png](tutorial_2_ray_tracing_files/tutorial_2_ray_tracing_15_0.png) + + + + + +![png](tutorial_2_ray_tracing_files/tutorial_2_ray_tracing_15_1.png) + + + +Mass profiles also have additional properties used in lensing calculations: + +- **convergence**: Represents the surface mass density of the profile in dimensionless units. +- **potential**: Represents the "lensing potential" of the mass profile in dimensionless units. +- **magnification**: Indicates how much brighter light rays appear due to the focusing effect of lensing. + +These quantities can be calculated using `*_from` methods and are returned as `Array2D` objects. + + +```python +convergence = sis_mass_profile.convergence_2d_from(grid=image_plane_grid) +potential_2d = sis_mass_profile.potential_2d_from(grid=image_plane_grid) +magnification_2d = al.LensCalc.from_mass_obj( + mass_obj=sis_mass_profile +).magnification_2d_from(grid=image_plane_grid) +``` + +The same plotter API can be used to visualize these properties: + + +```python +aplt.plot_array( + array=sis_mass_profile.convergence_2d_from(grid=image_plane_grid), + title="Convergence", +) +aplt.plot_array( + array=sis_mass_profile.potential_2d_from(grid=image_plane_grid), title="Potential" +) +``` + + + +![png](tutorial_2_ray_tracing_files/tutorial_2_ray_tracing_19_0.png) + + + + + +![png](tutorial_2_ray_tracing_files/tutorial_2_ray_tracing_19_1.png) + + + +One-dimensional plots can also be made using the same projection technique as in the previous tutorial: + + +```python +grid_2d_projected = image_plane_grid.grid_2d_radial_projected_from( + centre=sis_mass_profile.centre, angle=sis_mass_profile.angle() +) + +convergence_1d = sis_mass_profile.convergence_2d_from(grid=grid_2d_projected) + +plt.plot(grid_2d_projected[:, 1], convergence_1d) +plt.xlabel("Radius (arcseconds)") +plt.ylabel("Luminosity") +plt.show() +plt.close() +``` + + + +![png](tutorial_2_ray_tracing_files/tutorial_2_ray_tracing_21_0.png) + + + +The **convergence** and **potential** can be better understood when plotted in logarithmic space: + + +```python + +``` diff --git a/markdown/chapter_1_introduction/tutorial_2_ray_tracing_files/tutorial_2_ray_tracing_15_0.png b/markdown/chapter_1_introduction/tutorial_2_ray_tracing_files/tutorial_2_ray_tracing_15_0.png new file mode 100644 index 0000000000000000000000000000000000000000..2acd4b787f4a1e403a6b4ac4f87808235ea999d7 GIT binary patch literal 39428 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shows the example fully executed, with its real output images. +> Run it yourself via the [Python script](../../scripts/chapter_1_introduction/tutorial_3_more_ray_tracing.py) or the [Jupyter notebook](../../notebooks/chapter_1_introduction/tutorial_3_more_ray_tracing.ipynb). + +Tutorial 5: More Ray Tracing +============================ + +We'll now reinforce the ideas that we learnt about ray-tracing in the previous tutorial and introduce the following +new concepts: + +- What critical curves and caustics are. + - That by specifying redshifts and a cosmology, the results are converted from arc-second coordinates to physical + units of kiloparsecs (kpc). Again, if you're not an Astronomer, you may not be familiar with the unit of parsec, it + may be worth a quick Google! + - That a `Tracer` can be given any number of galaxies. + +Up to now, the planes have also had just one lens galaxy or source galaxy at a time. In this example, the tracer will +have multiple galaxies at each redshift, meaning that each plane has more than one galaxy. In terms of lensing +calculations: + +- If two or more lens galaxies are at the same redshift in the image-plane, the convergences, potentials and +deflection angles of their mass profiles are summed when performing lensing calculations. + +- If two or more source galaxies are at the same redshift in the source-plane, their light can simply be summed before +ray tracing. + +The `Tracer` fully accounts for this. + +__Contents__ + +- **Initial Setup:** To begin, lets setup the grid we'll ray-trace using. +- **Concise Code:** Lets set up the tracer used in the previous tutorial. +- **Critical Curves:** To end, we can finally explain what the black lines that have appeared on many of the plots. +- **Caustics:** In the previous tutorial, we plotted the critical curves of the mass profile on the image-plane. +- **Units:** Lets plot the lensing quantities again. +- **More Complexity:** We now make a lens with some attributes we didn`t in the last tutorial. +- **Multi Galaxy Ray Tracing:** Now lets pass our 4 galaxies to a `Tracer`, which means the following will occur. +- **Wrap Up:** Summary of the script and next steps. + + +```python + +from autoconf import jax_wrapper # Sets JAX environment before other imports + +from autoconf import setup_notebook; setup_notebook() + +import numpy as np +import autolens as al +import autolens.plot as aplt +``` + + Working Directory has been set to `HowToLens` + + +__Initial Setup__ + +To begin, lets setup the grid we'll ray-trace using. But, lets do something crazy and use a higher resolution than +the previous tutorials! + +Lets also stop calling it the `image_plane_grid`, and just remember from now on our `grid` is in the image-plane. + + +```python +grid = al.Grid2D.uniform(shape_native=(250, 250), pixel_scales=0.02) +``` + +The grid is now shape 250 x 250, which has more image-pixels than the 100 x 100 grid used previously. + + +```python +print(grid.shape_native) +print(grid.shape_slim) +``` + + (250, 250) + 62500 + + +__Concise Code__ + +Lets set up the tracer used in the previous tutorial. + +Up to now, we have set up each profile one line at a time, making the code long and cumbersome to read. + +From here on, we'll set up galaxies in a single block of code, making it more concise and readable. + + +```python +lens = al.Galaxy( + redshift=0.5, mass=al.mp.IsothermalSph(centre=(0.0, 0.0), einstein_radius=1.6) +) + +source = al.Galaxy( + redshift=1.0, + light=al.lp.SersicCoreSph( + centre=(0.0, 0.0), + intensity=1.0, + effective_radius=1.0, + sersic_index=1.0, + radius_break=0.025, + ), +) + +tracer = al.Tracer(galaxies=[lens, source]) +``` + +__Critical Curves__ + +To end, we can finally explain what the black lines that have appeared on many of the plots throughout this chapter +actually are. + +These lines are called the 'critical curves', and they define line of infinite magnification due to a mass profile. +They therefore mark where in the image-plane a mass profile perfectly `focuses` light rays such that if a source is +located there, it will appear very bright: potentially 10-100x as brighter than its intrinsic luminosity. + +The outer white line is a `tangential_critical_curve`, because it describes how the image of the source galaxy is stretched +tangentially. There is also an inner `radial_critical_curve` which appears in yellow on figures, which describes how the +image of the source galaxy is stretched radially. + +However, a radial critical curve only appears when the lens galaxy's mass profile is shallower than isothermal (e.g. +when its inner mass slope is less steep than a steep power-law). To make it appear below, we therefore change +the mass profile of our lens galaxy to a `PowerLawSph` with a slope of 1.8. + +In the next tutorial, we'll introduce 'caustics', which are where the critical curves map too in the source-plane. + + +```python +mass_profile = al.mp.PowerLawSph(centre=(0.0, 0.0), einstein_radius=1.6, slope=1.8) + +lens = al.Galaxy(redshift=0.5, mass=mass_profile) + +tangential_critical_curve_list = al.LensCalc.from_mass_obj( + mass_obj=mass_profile +).tangential_critical_curve_list_from(grid=grid) +radial_critical_curves_list = al.LensCalc.from_mass_obj( + mass_obj=mass_profile +).radial_critical_curve_list_from(grid=grid) + +``` + +__Caustics__ + +In the previous tutorial, we plotted the critical curves of the mass profile on the image-plane. We will now plot the +'caustics', which correspond to each critical curve ray-traced to the source-plane. This is computed by using the +lens galaxy mass profile's to calculate the deflection angles at the critical curves and ray-trace them to the +source-plane. + +As discussed in the previous tutorial, critical curves mark regions of infinite magnification. Thus, if a source +appears near a caustic in the source plane it will appear significantly brighter than its true luminosity. + +We again have to use a mass profile with a slope below 2.0 to ensure a radial critical curve and therefore radial +caustic is formed. As above, the tangential critical curve is white and maps to the tangential caustic in the source-plane, +which is also white. The radial critical curve is yellow and maps to the radial caustic, which is also yellow. + +We can plot both the tangential and radial critical curves and caustics using the `lines=`/`positions=` overlays object. The +critical curves are plotted only for the image-plane grid, whereas the caustic in the source plane. + + +```python +sis_mass_profile = al.mp.IsothermalSph(centre=(0.0, 0.0), einstein_radius=1.6) + +tracer = al.Tracer(galaxies=[lens, source]) + +tangential_critical_curve_list = al.LensCalc.from_tracer( + tracer=tracer +).tangential_critical_curve_list_from(grid=grid) +radial_critical_curves_list = al.LensCalc.from_tracer( + tracer=tracer +).radial_critical_curve_list_from(grid=grid) + + +traced_grid_list = tracer.traced_grid_2d_list_from(grid=grid) +source_plane_grid = np.asarray(traced_grid_list[1]) +source_plane_grid = source_plane_grid[np.isfinite(source_plane_grid).all(axis=1)] + +aplt.plot_grid(grid=traced_grid_list[0], title="Plane 0 Grid") +aplt.plot_grid( + grid=al.Grid2DIrregular(values=source_plane_grid), + title="Plane 1 Grid", +) + +tangential_caustic_list = al.LensCalc.from_tracer( + tracer=tracer +).tangential_caustic_list_from(grid=grid) +radial_caustics_list = al.LensCalc.from_tracer(tracer=tracer).radial_caustic_list_from( + grid=grid +) + +``` + + + +![png](tutorial_3_more_ray_tracing_files/tutorial_3_more_ray_tracing_11_0.png) + + + + + +![png](tutorial_3_more_ray_tracing_files/tutorial_3_more_ray_tracing_11_1.png) + + + +We can also plot the caustic on the source-plane image. + + +```python +aplt.plot_array(array=tracer.image_2d_list_from(grid=grid)[1], title="Plane 1 Image") +``` + + + +![png](tutorial_3_more_ray_tracing_files/tutorial_3_more_ray_tracing_13_0.png) + + + +Caustics also mark the regions in the source-plane where the multiplicity of the strong lens changes. That is, +if a source crosses a caustic, it goes from 2 images to 1 image. Try and show this yourself by changing the (y,x) +centre of the source-plane galaxy's light profile! + + +```python +source = al.Galaxy( + redshift=1.0, + light=al.lp.SersicCoreSph( + centre=(0.0, 0.0), + intensity=1.0, + effective_radius=1.0, + sersic_index=1.0, + ), +) + +tracer = al.Tracer(galaxies=[lens, source]) + +aplt.plot_array(array=tracer.image_2d_list_from(grid=grid)[1], title="Plane 1 Image") +``` + + + +![png](tutorial_3_more_ray_tracing_files/tutorial_3_more_ray_tracing_15_0.png) + + + +__Units__ + +Lets plot the lensing quantities again. However, we'll now use the `Units` object of the **PyAutoLens** plotter module +to set `in_kpc=True` and therefore plot the y and x axes in kiloparsecs. + +This conversion is performed automatically, using the galaxy redshifts and cosmology. + + +```python + +aplt.subplot_tracer(tracer=tracer, grid=grid) +aplt.subplot_galaxies_images(tracer=tracer, grid=grid) +``` + + + +![png](tutorial_3_more_ray_tracing_files/tutorial_3_more_ray_tracing_17_0.png) + + + + + +![png](tutorial_3_more_ray_tracing_files/tutorial_3_more_ray_tracing_17_1.png) + + + +If you're too familiar with Cosmology, it will be unclear how exactly we converted the distance units from +arcseconds to kiloparsecs. You'll need to read up on your Cosmology lecture to understand this properly. + +You can create a `Cosmology` object, which provides many methods for calculation different cosmological quantities, +which are shown below (if you're not too familiar with cosmology don't worry that you don't know what these mean, +it isn't massively important for using **PyAutoLens**). + +We will use a flat lambda CDM cosmology, which is the standard cosmological model often assumed in scientific studies. + + +```python +cosmology = al.cosmo.FlatLambdaCDM(H0=70, Om0=0.3) + +print("Image-plane arcsec-per-kpc:") +print(cosmology.arcsec_per_kpc_from(redshift=0.5)) +print("Image-plane kpc-per-arcsec:") +print(cosmology.kpc_per_arcsec_from(redshift=0.5)) +print("Angular Diameter Distance to Image-plane (kpc):") +print(cosmology.angular_diameter_distance_to_earth_in_kpc_from(redshift=0.5)) + +print("Source-plane arcsec-per-kpc:") +print(cosmology.arcsec_per_kpc_from(redshift=1.0)) +print("Source-plane kpc-per-arcsec:") +print(cosmology.kpc_per_arcsec_from(redshift=1.0)) +print("Angular Diameter Distance to Source-plane:") +print(cosmology.angular_diameter_distance_to_earth_in_kpc_from(redshift=1.0)) + +print("Angular Diameter Distance From Image To Source Plane:") +print( + cosmology.angular_diameter_distance_between_redshifts_in_kpc_from( + redshift_0=0.5, redshift_1=1.0 + ) +) +print("Lensing Critical convergence:") +print( + cosmology.critical_surface_density_between_redshifts_solar_mass_per_kpc2_from( + redshift_0=0.5, redshift_1=1.0 + ) +) +``` + + Image-plane arcsec-per-kpc: + 0.1638289752817242 + Image-plane kpc-per-arcsec: + 6.103926355398221 + Angular Diameter Distance to Image-plane (kpc): + 1259025.187042171 + Source-plane arcsec-per-kpc: + 0.12487552857882515 + Source-plane kpc-per-arcsec: + 8.007974111346966 + Angular Diameter Distance to Source-plane: + 1651763.228507974 + Angular Diameter Distance From Image To Source Plane: + 707494.3382263458 + Lensing Critical convergence: + 3083526795.6972556 + + +__More Complexity__ + +We now make a lens with some attributes we didn`t in the last tutorial: + + - A light profile representing a `bulge` of stars, meaning the lens galaxy's light will appear in the image for the + first time. + - An external shear, which accounts for the deflection of light due to line-of-sight structures. + + +```python +lens = al.Galaxy( + redshift=0.5, + bulge=al.lp.SersicSph( + centre=(0.0, 0.0), intensity=2.0, effective_radius=0.5, sersic_index=2.5 + ), + mass=al.mp.Isothermal( + centre=(0.0, 0.0), ell_comps=(0.0, -0.111111), einstein_radius=1.6 + ), + shear=al.mp.ExternalShear(gamma_1=0.05, gamma_2=0.0), +) + +print(lens) +``` + + Redshift: 0.5 + Light Profiles: + SersicSph + centre: (0.0, 0.0) + ell_comps: (0.0, 0.0) + intensity: 2.0 + effective_radius: 0.5 + sersic_index: 2.5 + Mass Profiles: + Isothermal + centre: (0.0, 0.0) + ell_comps: (0.0, -0.111111) + einstein_radius: 1.6 + slope: 2.0 + core_radius: 0.0 + ExternalShear + centre: (0.0, 0.0) + ell_comps: (0.0, 0.0) + gamma_1: 0.05 + gamma_2: 0.0 + + +Lets also create a small satellite galaxy nearby the lens galaxy and at the same redshift. + + +```python +lens_satellite = al.Galaxy( + redshift=0.5, + bulge=al.lp.DevVaucouleursSph( + centre=(1.0, 0.0), intensity=2.0, effective_radius=0.2 + ), + mass=al.mp.IsothermalSph(centre=(1.0, 0.0), einstein_radius=0.4), +) + +print(lens_satellite) +``` + + Redshift: 0.5 + Light Profiles: + DevVaucouleursSph + centre: (1.0, 0.0) + ell_comps: (0.0, 0.0) + intensity: 2.0 + effective_radius: 0.2 + sersic_index: 4.0 + Mass Profiles: + IsothermalSph + centre: (1.0, 0.0) + ell_comps: (0.0, 0.0) + einstein_radius: 0.4 + slope: 2.0 + core_radius: 0.0 + + +Lets have a quick look at the appearance of our lens galaxy and its satellite. + + +```python + +``` + +And their deflection angles, noting that the satellite does not contribute as much to the deflections. + + +```python + + +# NOTE: In the new API, pass title directly as a string: +# title="Lens Galaxy Deflections (x)" + +``` + +Now, lets make two source galaxies at redshift 1.0. Instead of using the name `light` for the light profiles, lets +instead use more descriptive names that indicate what morphological component of the galaxy the light profile +represents. In this case, we'll use the terms `bulge` and `disk`, the two main structures that a galaxy can be made of + + +```python +source_0 = al.Galaxy( + redshift=1.0, + bulge=al.lp.DevVaucouleursSph( + centre=(0.1, 0.2), intensity=0.3, effective_radius=0.3 + ), + disk=al.lp.ExponentialCore( + centre=(0.1, 0.2), + ell_comps=(0.111111, 0.0), + intensity=3.0, + effective_radius=2.0, + ), +) + +source_1 = al.Galaxy( + redshift=1.0, + disk=al.lp.ExponentialCore( + centre=(-0.3, -0.5), + ell_comps=(0.1, 0.0), + intensity=8.0, + effective_radius=1.0, + ), +) + +print(source_0) +print(source_1) +``` + + Redshift: 1.0 + Light Profiles: + DevVaucouleursSph + centre: (0.1, 0.2) + ell_comps: (0.0, 0.0) + intensity: 0.3 + effective_radius: 0.3 + sersic_index: 4.0 + ExponentialCore + centre: (0.1, 0.2) + ell_comps: (0.111111, 0.0) + intensity: 3.0 + effective_radius: 2.0 + sersic_index: 1.0 + radius_break: 0.01 + alpha: 3.0 + gamma: 0.25 + Redshift: 1.0 + Light Profiles: + ExponentialCore + centre: (-0.3, -0.5) + ell_comps: (0.1, 0.0) + intensity: 8.0 + effective_radius: 1.0 + sersic_index: 1.0 + radius_break: 0.01 + alpha: 3.0 + gamma: 0.25 + + +Lets look at our source galaxies (before lensing) + + +```python + +``` + +__Multi Galaxy Ray Tracing__ + +Now lets pass our 4 galaxies to a `Tracer`, which means the following will occur: + + - Using the galaxy redshift`s, and image-plane and source-plane will be created each with two galaxies galaxies. + +We've also pass the tracer below a Planck15 cosmology, where the cosomology of the Universe describes exactly how +ray-tracing is performed. + + +```python +tracer = al.Tracer( + galaxies=[lens, lens_satellite, source_0, source_1], + cosmology=al.cosmo.Planck15(), +) +``` + +We can now plot the tracer`s image, which now there are two galaxies in each plane is computed as follows: + + 1) First, using the image-plane grid, the images of the lens galaxy and its satellite are computed. + + 2) Using the mass profiles of the lens and its satellite, their deflection angles are computed. + + 3) These deflection angles are summed, such that the deflection of light due to the mass profiles of both galaxies in + the image-plane is accounted for. + + 4) These deflection angles are used to trace every image-grid coordinate to the source-plane. + + 5) The image of the source galaxies is computed by summing both of their images and ray-tracing their light back to + the image-plane. + +This process is pretty much the same as we have single in previous tutorials when there is one galaxy per plane. We +are simply summing the images and deflection angles of the galaxies before using them to perform ray-tracing. + + +```python +aplt.plot_array(array=tracer.image_2d_from(grid=grid), title="Image") +``` + + + +![png](tutorial_3_more_ray_tracing_files/tutorial_3_more_ray_tracing_35_0.png) + + + +As we did previously, we can plot the source plane grid to see how each coordinate was traced. + + +```python +aplt.plot_grid(grid=tracer.traced_grid_2d_list_from(grid=grid)[1], title="Plane 1 Grid") +``` + + + +![png](tutorial_3_more_ray_tracing_files/tutorial_3_more_ray_tracing_37_0.png) + + + +We can zoom in on the source-plane to reveal the inner structure of the caustic. + + +```python + +aplt.plot_grid(grid=tracer.traced_grid_2d_list_from(grid=grid)[1], title="Plane 1 Grid") +``` + + + +![png](tutorial_3_more_ray_tracing_files/tutorial_3_more_ray_tracing_39_0.png) + + + +__Wrap Up__ + +Tutorial 6 completed! Try the following: + + 1) If you change the lens and source galaxy redshifts, does the tracer's image change? + + 2) What happens to the cosmological quantities as you change these redshifts? Do you remember enough of your + cosmology lectures to predict how quantities like the angular diameter distance change as a function of redshift? + + 3) The tracer has a small delay in being computed, whereas other tracers were almost instant. What do you think + is the cause of this slow-down? + + +```python + +``` diff --git a/markdown/chapter_1_introduction/tutorial_3_more_ray_tracing_files/tutorial_3_more_ray_tracing_11_0.png b/markdown/chapter_1_introduction/tutorial_3_more_ray_tracing_files/tutorial_3_more_ray_tracing_11_0.png new file mode 100644 index 0000000000000000000000000000000000000000..96ec34a77a83c63a556542d441d731e1ec57e9a2 GIT binary patch literal 27026 zcmeHw2UL{jx;3_7#{`w5C>TYsL`ak(AfQH3q^UIN8kNpahF*;tjRgrHB3-3-5h+7e zi2^FU!vG>k8yHj=kpAzNb6diAPIAkC*Ig?vCm3etD{pzqe)hA!YsXcNtXR5!DF+9~ z3WcMGPI7R}x5dk6@fY|_#q=Q>en{CJK4qt7Wn|}g#`Zji@)HT)z*2pchL&uzM)}pLLn!JY3FxYk}Y#{N&*1#_w@=ntjd$_M1db)dF z;*)C|_D5Pcz13)rNK90caUH3tJi?!(OQ9Y1_xCT=cfN79G)OFawAI!AQ~T9~Bi?*w zE!pYW4g>K&F5v8QESLz*pZxeru=fM2-jI>vIh9Z*#rTFb*k&l*|L{`NW7CMFjHY8P z+4j{J<}DZ=3ZKz)q}yCwD`G$w2<3Rf&B<}%E2izwe)7%F8J2K*f11@-tEhPAo659u*#xVC 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