Speed of orbiting planet given eccentricity of orbit

In summary, the eccentricity of a planet's orbit about the sun is 0.4. To find the ratio of the lengths of the major to minor axes of the orbit, we use the formula a/b = 1/sqrt(1 - e^2), which gives us a/b = 1.0911. To find the ratio of the speeds of the planet at the farthest and closest points in its orbit, we use the formula v = sqrt(mu * (2/r - 1/a)), where mu is the standard gravitational parameter and r is the distance from the sun. Using conservation of angular momentum, we can simplify this and get the correct answer.
  • #1
ephedyn
170
1

Homework Statement


If the eccentricity of a planet's orbit about the sun is 0.4, find (a) the ratio of the lengths of the major to minor axes of the planet's orbit, and (b) the ratio of the speeds of the planet when it is at the ends of the major axis of its elliptical orbit.


Homework Equations



[itex]E=\dfrac{1}{2} mv^2 + \dfrac{-GMm}{r} + \dfrac{L^2}{2mr^2}[/itex]

The Attempt at a Solution



The first part is rather short and sweet:

[itex]\dfrac{a}{b}=\dfrac{1}{\sqrt{1-\epsilon^{2}}}=\dfrac{1}{\sqrt{1-0.4^{2}}}\approx1.0911[/itex]

But I have no idea how to proceed on the second part. If I take the ratio of the kinetic energies I get a very messy expression and have problem eliminating the total energy E. Is there a more elegant way to do this?
 
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  • #2
hi

orbital speed is given by

[tex]v=\sqrt{\mu\left(\frac{2}{r}-\frac{1}{a}\right)}[/tex]

where [tex]\inline{\mu =G(M+m)}[/tex] is the standard gravitational parameter. and r is the distance from the sun(focus). at the farthest point in the orbit from the sun
[tex]\inline{r=(1+e)a}[/tex] and the closest point we have [tex]\inline{r=(1-e)a}[/tex]
where a is semi major axis length. use this info

Newton
 
Last edited:
  • #3
Hey thanks for your help. I tried using your approach but I couldn't simplify it so I tried using conservation of angular momentum and got the correct answer. Simpler than I expected.
 
  • #4
well algebra is pretty straight forward I guess.
 
  • #5




To find the ratio of the speeds of the planet at the ends of the major axis, we can use the fact that the angular momentum (L) is conserved in an elliptical orbit. This means that at each point in the orbit, the product of the distance (r) and the speed (v) will be constant. Therefore, we can set up the following equation:

rv=constant

At the ends of the major axis, the distance (r) will be equal to the length of the major axis (a), and the speed (v) will be the fastest in the orbit. Therefore, at this point, we can say:

a_{max}v_{max}=constant

Similarly, at the ends of the minor axis, the distance (r) will be equal to the length of the minor axis (b), and the speed (v) will be the slowest in the orbit. Therefore, at this point, we can say:

b_{max}v_{min}=constant

Dividing these two equations, we get:

\dfrac{a_{max}}{b_{max}}\dfrac{v_{max}}{v_{min}}=1

Using the ratio of the major to minor axes found earlier, we can substitute and solve for the ratio of the speeds:

\dfrac{v_{max}}{v_{min}}=\dfrac{b_{max}}{a_{max}}=\dfrac{1}{1.0911}\approx0.917

Therefore, the ratio of the speeds at the ends of the major axis is approximately 0.917. This means that the planet will be moving at about 91.7% of its maximum speed when it is at the ends of the major axis of its orbit.
 

Related to Speed of orbiting planet given eccentricity of orbit

1. What is the definition of eccentricity in relation to the orbit of a planet?

Eccentricity is a measure of how elliptical, or non-circular, an orbit is. It is the ratio of the distance between the foci of the ellipse to the major axis of the ellipse.

2. How does eccentricity affect the speed of an orbiting planet?

The speed of an orbiting planet is affected by its eccentricity because a more elliptical orbit means that the planet will be closer to the center of its orbit at some points and further away at others. This results in the planet moving at varying speeds as it orbits.

3. Is there a direct relationship between eccentricity and the speed of an orbiting planet?

Yes, there is a direct relationship between eccentricity and the speed of an orbiting planet. The higher the eccentricity, the more variation in speed the planet will experience during its orbit.

4. How can the speed of an orbiting planet be calculated based on its eccentricity?

The speed of an orbiting planet can be calculated using Kepler's Third Law, which states that the square of the orbital period is proportional to the cube of the semi-major axis of the orbit. From this equation, the speed of the planet can be determined based on its eccentricity and the distance from the planet to the center of its orbit.

5. Can the speed of an orbiting planet change if its eccentricity changes?

Yes, the speed of an orbiting planet can change if its eccentricity changes. As the eccentricity of an orbit increases, the distance between the planet and the center of its orbit will vary more, resulting in changes in the planet's speed throughout its orbit.

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