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Gravitational stability of a planet within its rings

Developed by Michael Massa - Published May 3, 2023

This set of exercises uses a simplified picture of Saturn and its Rings to look at the gravitational force due to a continuous mass distribution acting on a point mass. Students will apply numerical integration to calculate the net force on the point mass at different positions, then perform a second integration to compute the gravitational potential energy and assess the gravitational stability of the planet when centered within its ring. Saturn's ring is then replaced by a thin spherical shell, which results in an additional example of stability, as well as demonstrating the gravitational equivalent of Gauss' Law in electrostatics.
Subject Areas Mechanics and Astronomy Beyond the First Year Python Students who complete this exercise set should be able to: - Set up integral expressions for the gravitational force of a continuous mass distribution in a dimensionless form (Exercise 1) - Use simple numerical methods to solve definite integrals, verify these methods with analytical solutions, and extend to integrals for which there are no closed-form solutions (Exercises 2 - 5) - Numerically integrate the gravitational force along a path to obtain the potential energy, $U(\vec{r})$, relative to a reference position (Exercises 3 - 5) - Plot the gravitational force, $F(\vec{r})$, and potential energy, $U(\vec{r})$, and use both representations to assess the gravitational stability of the planet at the centre of a continuous mass distribution (Exercises 2 - 5) 180 min

These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).

Consider the gravitational attraction between *Saturn* and its *Rings*. In reality, the planet remains centered within the rings, but consider an (over-)simplified model, which treats the *Rings* as a single solid ring with uniform density, and compute the gravitational force experienced by the planet at different positions within the plane of the ring. ## Exercise 1: Gravitational force due to a uniform circular ring Using ink & paper, set up the integral to calculate the net gravitational force experienced by the planet due to the rings. - Situate the ring in the xz-plane, centred on the origin. - Treat the *Rings* as a single, stationary, thin circular ring of mass $M$ and radius $R$, with uniform linear density $\lambda = M/2\pi R$ - Treat *Saturn* as a point mass, $m$, located on the z-axis at distance $z < R$. - By symmetry, the force, $F(z)$ will point along the z-axis; it is sufficient to express the integrand in terms of the z-component of the force, $dF_z$. This integral cannot be solved analytically; you will numerically compute the result in the next exercise. However, with computational work, it is often worth recasting the integral into dimensionless quantities, rather than retaining units. Express the position coordinate in terms of a dimensionless distance, $\xi = z/R$, and scale the net force by a representative value for this system, $F^* = \frac{GmM}{R^2}$. $$f_{ring}(\xi) = \frac{F(\xi)}{F^*} = \frac{1}{\pi} \int_{0}^{\pi} \frac{\cos(\theta)-\xi}{[1 + \xi^2 - 2\xi\cos(\theta)]^{3/2}} d\theta$$ Note: The symmetry of the integrand allows for the simplification of integration limits: $\int_{0}^{2\pi} \rightarrow 2\int_{0}^{\pi}$. ## Exercise 2: Numerical integration of gravitational force Equation (1) can be numerically integrated using a pre-existing library/package, or using simple numerical methods such as *Trapezoidal Rule* or *Simpson's Rule*. 1. Create a function, *integrand*, which takes inputs $\theta$ and $\xi$, and returns the corresponding value of the integrand. 2. Plot the integrand as a function of $\theta$ for various planet-positions, $\xi$, to get a sense of the gravitational force from different parts of the ring. 3. Numerically integrate equation (1) for a range of planet positions, $\xi \in [0,0.9]$, and plot the gravitational force vs position. ## Exercise 3: Gravitational potential energy and stability The potential energy can be obtained by integration: $$U(P) = -\int_\mathcal{O}^P \overrightarrow{f}\cdot \overrightarrow{dr}$$ 1. Use the results from the previous exercise to numerically integrate $f_{ring}(\xi)$ to obtain the potential energy at each position, $\xi$. 2. Plot the gravitational potential energy vs position, using the same range for $\xi$ as in the previous exercise. **Note:** The reference position, $\mathcal{O}$, is typically chosen to be infinitely far away, where the potential energy is assumed to be zero. However, for this exercise the origin is a useful reference position. ## Exercise 4: Gravitational force and energy out of the plane Exercises 1-3 can be repeated to compute the force and potential energy for planet-positions *out of the plane of the ring*. For simplicity, consider positions along the y-axis (i.e. the axis of the ring). 1. Using ink & paper, set up the integral to calculate the net gravitational force. Work with dimensionless position $\zeta = y/R$ and scale the force by the same value, $F^*$. 2. This integrals to obtain the force, $f_{ring}(\zeta)$, and the gravitational potential energy, $U_{ring}(\zeta)$, can both be solved analytically, or could be done numerically using the methods in Exercises 2 & 3. 3. Add plots of $f_{ring}(\zeta)$ and $U_{ring}(\zeta)$ to your results for the in-plane position of the planet. How does the gravitational stability compare for in- vs out-of-plane positions? ## Exercise 5: Gravitational force due to a uniformly dense spherical shell Next, consider the planet to be surrounded by a thin spherical shell of mass $M$ and radius $R$, with a uniform density $\sigma = M/4\pi R^2$. Calculate the gravitational force experienced by the planet for positions within the shell. Based on the symmetry, it is enough to consider positions along the z-axis, as in Exercise 1. While the mass is distributed over a 2D surface, the force can still be calculated using a 1D integral. In equation (1) the infinitesimal mass element is a segment of arc length, $dM = \lambda Rd\theta$, where $\lambda$ is the linear mass density of the ring. For the spherical shell, which has area mass density $\sigma$, treat the infinitesimal unit as a ring, where $$dM = \sigma R\sin(\theta) d\theta d\phi = \sigma 2\pi \sin(\theta) R d\theta = \lambda(\theta) R d\theta.$$ 1. Modify your function, *integrand*, from Exercise 2 for the case of a spherical shell. 2. Plot the integrand as a function of $\theta$ for various planet-positions, $\xi$. How does the $\theta$-dependence of the force due to the shell compare with that of the ring? 3. Numerically integrate to solve for the force, and add the $f_{shell}(\xi)$ curve to you plot of force vs position. 4. Compute $U_{shell}(\xi)$ by numerically integrating $f_{shell}(\xi)$, and add this to your energy plot.

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Credits and Licensing

Michael Massa, "Gravitational stability of a planet within its rings," Published in the PICUP Collection, May 2023, https://doi.org/10.1119/PICUP.Exercise.gravstability.

The instructor materials are ©2023 Michael Massa.

The exercises are released under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 license