Spacecraft orbiting sun, change in velocity

[karnten07]

Homework Statement

A spacecraft is initially in a circular orbit of the sun at the Earth's orbital radius. It uses a single brief rocket thrust parallel to it's velocity to put it in a new orbit with aphelion distance equal to the radius of Jupiter's orbit.

What is the ratio of the spacecraft 's speeds just after and just before the rocket thrust?

Assume Earth and Jupiter have cicular orbits, with radii 1.5 x 10^11m and 7.8x10^11m respectively. The mass of the sun is 1.99 x10^30kg and G = 6.672x10^-11 m^3kg^-1s^-2. Ignore gravitational attraction between the spacecraft and planets.

Homework Equations

The Attempt at a Solution

Using L = [r x mv]

I equate the angular momentum at Earth's orbit and at jupiters orbit.

1.5x10^11 * mV1 = 7.8 x10^11 *mV2

So the m's cancel and V2/V1 = 1.5/7.8

Now I am very unsure about this because i am asked for the ratio of velocities just before and after the thrust. But what i have calculated is the velocity of the spacecraft when it is in jupiters orbit. So this means that my V2 is slower than V1 but V2 just after the thrust is going to be faster than V1. Perhaps i just invert the fraction to show the change in velocity?

So i would get V2/V1 = 7.8/1.5

Any ideas guys? There is a second part to this question which is where i am to use G and the mass of the sun so i didnt think id need them for this part.

 

[D H]

Now I am very unsure about this because i am asked for the ratio of velocities just before and after the thrust. But what i have calculated is the velocity of the spacecraft when it is in jupiters orbit.

What you calculated is not what the question asked for. You were told the new orbit's apehelion distance equals the radius of Jupiter's orbit. What is the new orbit's perihelion distance? From that, you should be able to compute the new orbit's eccentricity and the velocity just after completing the burn.

 

[karnten07]

What you calculated is not what the question asked for. You were told the new orbit's apehelion distance equals the radius of Jupiter's orbit. What is the new orbit's perihelion distance? From that, you should be able to compute the new orbit's eccentricity and the velocity just after completing the burn.

I don't see how to calculate the perihelion for this orbit. Is the new orbit not circular as it was when it was at Earth's orbital radius, giving it 0 eccentricity?

 

[D H]

The satellite doesn't teleport to Jupiter's orbit. Instead, "it uses a single brief rocket thrust parallel to it's velocity to put it in a new orbit with aphelion distance equal to the radius of Jupiter's orbit."

The rocket undergoes some some non-zero change in velocity [itex]\Delta \vec v[/itex] in an extremely short period of time. By the mean value theorem, the position changes by [itex]\bar v \Delta t[/itex], where [itex]\bar v[/itex] is some value between [itex]\vec v[/itex] and [itex]\vec v + \Delta \vec v[/itex]. Since the rocket fires for "a brief instant", the position is essentially unchanged. (It is unchanged in the limit [itex]\Delta t \to 0[/itex])

Just after the end of the burn, the satellite is essentially at Earth's orbital radius. Some time later, it is at Jupiter's orbital radius without any intervening engine firings. This does not qualify as a circular orbit.

In a circular orbit, the velocity vector is always normal to the radial vector. Since the radial vector doesn't change (much) during the brief firing interval and since the thrust is parallel to the rocket's velocity vector, the post-burn velocity vector remains normal to the radial vector. At which points in an elliptical orbit is the velocity vector normal to the radial vector?

 

[karnten07]

What you calculated is not what the question asked for. You were told the new orbit's apehelion distance equals the radius of Jupiter's orbit. What is the new orbit's perihelion distance? From that, you should be able to compute the new orbit's eccentricity and the velocity just after completing the burn.

The new perihelion distance is the boost point i believe. I say this because it says it is like that in a similar question, but I am unsure if it is specific to that case or if physics determines that the perihelion would always occur at the boost point (or if the point moves, the distance from the focus stays at the boost radius). Anyone know?

 

[D H]

The post-burn velocity vector is normal to the position vector as a consequence of

• the pre-burn velocity vector is normal to the position vector (this is always true for a circular orbit),
• the post-burn velocity vector is parallel to the pre-burn velocity vector (given), and
• the burn is essentially instantaneous (given).

There are only two places in an elliptical orbit where the velocity vector is normal to the position vector: apofocus and perifocus. Thanks to the delta-V the satellite is on its way to Jupiter. The satellite is therefore at perifocus.

Note: This problem illustrates what is called a "http://liftoff.msfc.nasa.gov/academy/rocket_sci/satellites/hohmann.html"", the most energy-efficient means of transfering from one circular orbit to another in a simple gravitational system.