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Transcribed Image Text:Planetary motion.
A planet or asteroid orbits the Sun in an elliptical path'. The diagram shows two points of interest:
point P, the perihelion, where the planet is nearest to the Sun, and the far point, or aphelion, at point A.
At these two points (and only these two points), the velocity v of the planet is perpendicular to the
corresponding "radius" vector 7, from the Sun to the planet. An arbitrary point Q is also shown.
Suppose the asteroid's speed at aphelion is vą = 900 m/s, and it's distance from the sun there is ra =
35 AU (AU is the "astronomical unit", commonly used in planetary astronomy).
a) Determine the planet's speed vp at perihelion.
b) Determine the planet's distance rp from the Sun (in AU) at perihelion.
Assume that the Sun is orders of magnitude more massive than the asteroid. This means the Sun does
not accelerate significantly in response to the planet's gravity, and can be assumed to remain at rest.
Derive a symbolic formula, but don't try too hard to simplify it. Look up the values and conversion
factors you need in order to obtain a numerical answer.
Recall the universal gravity formulas:
-mMG
Force
r2
-mMG
Potential energy
Ug
Consider the Sun-asteroid system to be isolated and apply conservation of energy.
Also think about the system of the asteroid by itself. What influences are acting on the asteroid, and
how do these affect its energy, momentum, and/or angular momentum?
How can you write the asteroid's angular momentum (with respect to the Sun) at points A and P?
Approximately. The influence of other planets, while small, complicates the situation. In addition, Newton's law
of gravitation fails under conditions of very intense gravitation; this effect is small but relevant in analysis of
planetary systems.
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