New orbit of satellite deflected by Jupiter

[Dazed&Confused]

Homework Statement

This question involves parts of other questions, so I will state the relevant parts and essentially what the question is asking. It is from Kibble Classical Mechanics, Chapter 4, Question 22. If more information is needed I will provide.

We have a satellite that orbits the sun initially. It's orbit extends from Earth's to Jupiter's and it just touches both. Thus the speed of the satellite when near Jupiter is 7.4km/s while that of Jupiter is 13.1km/s. The satellite is deflected by Jupiter by 90 degrees in Jupiter's frame of reference so that in this frame the satellite ends up traveling away from the Sun. Thus in the frame of reference of the Sun the satellite is now traveling at 13.1km/s tangentially and 5.7km/s radially. The question asks you to find the new aphelion, perihelion, and orbital period.

Homework Equations

$$\tfrac12 m \dot{r}^2 + \frac{J^2}{2mr^2} + V(r) = E$$

The Attempt at a Solution

To me it appears that the energy of the satellite is now positive: the energy of Jupiter was zero (assuming a circular orbit) and now the satellite has the same tangential speed, same potential, but also extra radial speed. Thus the new orbit should be a hyperbola.

 

[gneill]

To me it appears that the energy of the satellite is now positive: the energy of Jupiter was zero (assuming a circular orbit) and now the satellite has the same tangential speed, same potential, but also extra radial speed. Thus the new orbit should be a hyperbola.

Perhaps you were thinking of the eccentricity being zero for a circular orbit? Bound orbits (such as those for planets) have negative mechanical energy. Jupiter's specific mechanical energy is:
$$\xi = \frac{v^2}{2} - \frac{\mu}{r}$$
$$ ~~~= \frac{(13.1~km/s)^2}{2} - \frac{(1.327 \times 10^{20}~m^3s^{-2})}{778.3 \times 10^6~km} = -8.47 \times 10^7~J/kg$$

The borderline case of a parabolic orbit has zero mechanical energy.

[edit: fixed math error. Sorry about that!]

 

[Dazed&Confused]

I feel a bit silly about that. Thanks.