External satellite resonance

In summary, the conversation discusses working through a paper titled "Dynamics of Planetary Rings" by Goldreich and Tremaine. The focus is on expanding the potential of an external satellite using a double Fourier series. The coordinates of the satellite in epicyclic oscillation are given by $r = a(1 - ecos(\kappa t))$ and $\theta = \Omega t + \frac{2\Omega e}{\kappa} sin(\kappa t)$. The paper explains that to the lowest order in eccentricity, the first coefficient (m,0) is given by $U_{m,0} = \frac{4\pi}{\kappa}\int_0^{2\pi
  • #1
Kelly Lam
2
0
I'm working through a paper "Dynamics of Planetary Rings" by Goldreich and Tremaine (http://www.annualreviews.org/doi/pdf/10.1146/annurev.aa.20.090182.001341). I'm working through p.22 about expanding the potential of an external satellite as a double Fourier series. The external satellite is in epicyclic oscillation with coordinates given by $r = a(1 - ecos(\kappa t))$ and $\theta = \Omega t + \frac{2\Omega e}{\kappa} sin(\kappa t)$ (these can be found in p.2 of http://articles.adsabs.harvard.edu//full/1980ApJ...241..425G/0000426.000.html). $\kappa$ is the epicyclic frequency and $\Omega$ is the circular orbital frequency at $r=a$.

I'm struggling to see how to the lowest order in e the eccentricity of the satellite, the first coefficient (m,0) is given as that in the paper.
 
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  • #2
The paper states that to the lowest order in e, this coefficient is given by$$U_{m,0} = \frac{4\pi}{\kappa}\int_0^{2\pi} r^m e cos(\kappa t) dt = \frac{8\pi a^m e}{\kappa^2}$$I'm struggling to see how the paper gets this result.
 

1. What is external satellite resonance?

External satellite resonance is a phenomenon in which a small celestial body, such as a moon or satellite, orbits around a larger celestial body in a specific ratio to the larger body's orbit. This results in a synchronized pattern of orbits, known as a resonance, which can affect the stability and behavior of both bodies.

2. How does external satellite resonance occur?

External satellite resonance occurs when a small celestial body orbits a larger body at a specific ratio, such as 2:1 or 3:2. This means that for every one orbit of the larger body, the smaller body completes two or three orbits. This synchronized pattern is caused by the gravitational interactions between the two bodies.

3. What are the effects of external satellite resonance?

The effects of external satellite resonance can vary depending on the specific resonance ratio and the characteristics of the bodies involved. In some cases, it can lead to the destabilization of orbits and potential collisions between the two bodies. In other cases, it can result in the formation of gaps or rings in the larger body's orbit.

4. Can external satellite resonance be observed?

Yes, external satellite resonance can be observed in our own solar system. For example, the moons of Jupiter and Saturn exhibit various resonance patterns with their respective planets. These resonances can also be observed in other planetary systems, providing insights into the dynamics of celestial bodies.

5. How does external satellite resonance affect space exploration?

External satellite resonance can have significant effects on space exploration, particularly when it comes to spacecraft trajectories. Scientists must take into account the resonance patterns of celestial bodies when planning missions and navigating through space. In some cases, resonance can also be used to assist with spacecraft maneuvers, such as gravitational assists.

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