Exploring Comet's Aphelion: Limiting Eccentricity and Closest Approach

In summary, the conversation discusses the behavior of a comet passing through its aphelion at a fixed distance from the sun. By letting the angular momentum approach zero, the eccentricity of the orbit approaches 1 and the distance of closest approach approaches zero. Taking limits and substituting in equations can help understand this behavior.
  • #1
kala
21
0

Homework Statement



Consider a comet which passes through its aphelion at a distance rmax from the sun. Imagine that, keeping rmax fixed, we somehow make the angular momentum l smaller and smaller though not actually zero; that is we let l[tex]\rightarrow[/tex]0. Use equations c=l2/[tex]\gamma\ mu[/tex] and rmax=c/1-[tex]\epsilon[/tex], rmin=1+[tex]\epsilon[/tex]to show that in this limit the eccentricity, [tex]\epsilon[/tex] of the elliptical orbit approaches 1 and the distance of closest approach rmin approaches zero.

Homework Equations



the equations posted above look funny, but for c it should be c=l2/gamma*mu
rmax=c/1-epsilon and rmin= c/1+epsilon.

The Attempt at a Solution


Well, I know that I am going to need to take limits. what i tried was to substitute in the c for the rmax equation, but then i really didn't know where to go from there. any help would be great.
 
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  • #2
Well, physically what happened if you too l--> 0 What you reall did is take your orbital velocity to zero.

L = mV X R

You assumed aphelion, so R was fixed. Assuming mass went to zero would just make the commet float away. What wil happen is the commet will make a b-line straight for the sun.
 

1. What is the significance of a comet's aphelion?

The aphelion of a comet is the point in its orbit where it is farthest from the sun. This distance is important because it affects the comet's behavior and characteristics, such as the amount of sunlight it receives and the amount of material it releases as it approaches the sun.

2. How does a comet's eccentricity affect its closest approach to the sun?

The eccentricity of a comet's orbit is a measure of how elongated or circular it is. The more eccentric a comet's orbit, the closer it will come to the sun at its closest approach. Comets with high eccentricities may have very different closest approach distances from one orbit to the next.

3. What limits a comet's eccentricity?

The primary factor that limits a comet's eccentricity is its interaction with other celestial bodies, such as planets and moons. These interactions can either increase or decrease a comet's eccentricity, depending on the timing and placement of the encounters.

4. How do scientists study a comet's aphelion and eccentricity?

Scientists study a comet's aphelion and eccentricity by observing its behavior and characteristics during its orbit. They use telescopes and other instruments to measure the comet's position, velocity, and composition, and use this data to calculate its orbit and determine its aphelion and eccentricity.

5. Why is it important to study a comet's aphelion and eccentricity?

Studying a comet's aphelion and eccentricity can provide valuable insights into the formation and evolution of our solar system. It can also help scientists understand the behavior and potential hazards of comets, which can have significant impacts on Earth. Additionally, studying a comet's aphelion and eccentricity can contribute to our overall knowledge of celestial bodies and their orbits.

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