Question

Animate a planet's orbit. A planet's orbit around a star has the shape of an ellipse. The purpose of this exercise is to make an animation of the movement along the orbit. One should see a small disk, representing the planet, moving along an elliptic curve. An evolving solid line shows the development of the planet's orbit as the planet moves. The points \((x, y)\) along the ellipse are given by the expressions $$ x=a \cos (\omega t), \quad y=b \sin (\omega t) $$ where \(a\) is the semimajor axis of the ellipse, \(b\) is the semiminor axis, \(\omega\) is an angular velocity of the planet around the star, and \(t\) denotes time. One complete orbit corresponds to \(t \in[0,2 \pi / \omega] .\) Let us discretize time into time points \(t_{k}=k \Delta t\), where \(\Delta t=2 \pi /(\omega n) .\) Each frame in the movie corresponds to \((x, y)\) points along the curve with \(t\) values \(t_{0}, t_{1}, \ldots, t_{i}, i\) representing the frame number \((i=1, \ldots, n)\). Let the plot title of each frame display the planet's instantaneous velocity magnitude. This magnitude is the length of the velocity vector $$ \left(\frac{d x}{d t}, \frac{d y}{d t}\right)=(-\omega a \sin (\omega t), \omega b \cos (\omega t)) $$ which becomes \(\omega \sqrt{a^{2} \sin ^{2}(\omega t)+b^{2} \cos ^{2}(\omega t)}\) Implement the visualization of the planet's orbit using the method above. Run the special case of a circle and verify that the magnitude of the velocity remains constant as the planet moves. Name of program file: planet_orbit.py.

Short answer

Use Python with NumPy and Matplotlib to animate an ellipse representing the planet's orbit.

Step-by-step solution

Import Necessary Libraries

To create the animation, we first need to import the necessary libraries for mathematical computations and visualization. We will use NumPy for calculations and Matplotlib for plotting. ```python import numpy as np import matplotlib.pyplot as plt from matplotlib.animation import FuncAnimation ```

Define Parameters

Define the fixed parameters for the ellipse including the semi-major axis \(a\), the semi-minor axis \(b\), the angular velocity \(\omega\), and the number of frames \(n\). Choose these values based on your preferences.```pythona = 5 # Length of the semi-major axisb = 3 # Length of the semi-minor axisomega = 2 * np.pi # Angular velocityn = 100 # Number of frames```

Calculate Time Points

Discretize the time into specific points from 0 to \(2\pi/\omega\) using the equation provided. Calculate \(\Delta t = \frac{2\pi}{\omega n}\) and create an array of time points.```pythondelta_t = 2 * np.pi / (omega * n)time_points = np.array([k * delta_t for k in range(n)])```

Define Ellipse Function

Create a function to compute the position \((x, y)\) on the ellipse for a given time \(t\). This uses the ellipse equations provided.```pythondef ellipse_pos(t): x = a * np.cos(omega * t) y = b * np.sin(omega * t) return x, y```

Define Velocity Magnitude Function

Create a function to compute the velocity magnitude given a time \(t\). Use the expression \(\omega \sqrt{a^2 \sin^2(\omega t) + b^2 \cos^2(\omega t)}\).```pythondef velocity_magnitude(t): return omega * np.sqrt((a * np.sin(omega * t))**2 + (b * np.cos(omega * t))**2)```

Initialize Plot

Initialize the plot with Matplotlib. Set up the figure, axis limits, and plot the elliptical orbit. ```python fig, ax = plt.subplots() ax.set_xlim(-1.1 * a, 1.1 * a) ax.set_ylim(-1.1 * b, 1.1 * b) planet_line, = ax.plot([], [], 'b-', lw=2) planet_dot, = ax.plot([], [], 'ro') ```

Animation Update Function

Define the update function for the animation. Update the planet's position and the plot title with the velocity magnitude for each frame. ```python def update(frame): t = time_points[frame] x, y = ellipse_pos(t) planet_line.set_data(time_points[:frame], ellipse_pos(time_points[:frame])) planet_dot.set_data(x, y) ax.set_title(f'Velocity: {velocity_magnitude(t):.2f}') return planet_line, planet_dot ```

Run Animation

Use the `FuncAnimation` function to animate the plot over all the frames. Show or save the created animation. ```python ani = FuncAnimation(fig, update, frames=n, blit=True) plt.show() ```

Key concepts

Elliptical Orbits

Elliptical orbits are a fascinating aspect of celestial mechanics, where planets move around stars in elliptical paths. This is due to the gravitational forces that act between the star and the orbiting body. An ellipse has two axes: the major axis, which is the longest diameter, and the minor axis, which is the shortest. In our exercise, these are represented by the semimajor axis \(a\) and the semiminor axis \(b\), respectively.
In mathematical terms, the position of a planet on an elliptical orbit can be calculated using trigonometric functions that involve time \(t\):

• The x-coordinate is given by \(x = a \cos(\omega t)\).
• The y-coordinate is given by \(y = b \sin(\omega t)\).

The parameter \(\omega\) is the angular velocity, which determines the speed at which the planet travels along its path. The time variable \(t\) advances to simulate the planet's movement in the elliptical orbit, and one full orbit corresponds to \(t \in [0, 2\pi/\omega]\).
Understanding these parameters and equations is crucial for visualizing and computing planetary orbits accurately, as they depict the actual celestial motion.

Matplotlib

Matplotlib is a powerful Python library used extensively for plotting. It allows the creation of static, interactive, and animated visualizations across many different fields. In the context of our exercise, Matplotlib is utilized to animate the path of a planet orbiting its star along an ellipse. Here's how it works:

• First, we need to initialize a plot using the `figure` and `axes` objects. This sets up the canvas where we will create the orbit animation.


• The `plot` function is used to draw the ellipse that represents the planet's path, setting axis limits relative to the semimajor and semiminor axes defined earlier.


• Then, the `FuncAnimation` function from Matplotlib is employed to create the animation by updating the position of the planet and lines for each frame based on calculated positions over time.

Matplotlib is advantageous in this application due to its flexible APIs and integration with numerical libraries like NumPy. This allows for seamless calculation updates that are crucial for real-time animations. You can view the evolving orbit with dynamic title updates showing the instantaneous velocity, enhancing both understanding and visualization.

Velocity Computation

Calculating velocity is critical for understanding the dynamics of orbital motion. The velocity vector arises from the changes in position over time and, for an elliptical orbit, it varies as the planet moves. The velocity magnitude is calculated using derivatives of the parameterized ellipse functions for \(x\) and \(y\). The expressions for these derivatives are:

• The derivative of \(x\) with respect to \(t\) gives the velocity in the x-direction: \(\frac{dx}{dt} = -\omega a \sin(\omega t)\).
• The derivative of \(y\) with respect to \(t\) gives the velocity in the y-direction: \(\frac{dy}{dt} = \omega b \cos(\omega t)\).

The magnitude of the velocity vector is then given by the expression:\[u = \omega \sqrt{a^2 \sin^2(\omega t) + b^2 \cos^2(\omega t)}\]
This formula tells us that the velocity depends on the ellipse's dimensions and the angular speed, \(\omega\). By calculating and displaying this magnitude dynamically as the planet moves, we get insights into how speeds vary along different sections of the orbit, crucial for astrodynamics simulations and visualizations.