Neptune orbits the Sun (Msun = 1.99*1030 kg) with an average orbital radius of r = 30.1 Astronomical Units = 4.514*1012 m, and a typical velocity of 5.42 km/s. What velocity would be required for Neptune to leave the solar system from its current radius?
Two earth-like planets, each with mass m = 6*1024 kg, orbit each other in a circular orbit once every 50 days. Find the distance between their centers in km.
Consider a particle of mass m moving in the xy plane. The potential energy
function is
U(x,y) = (k/2)(x2 + y2), with k a positive constant.
(a) Find the equations of motion.
(b) Are there circular orbits? If yes, do they all have the same period?
(c) Is the total energy conserved?
A satellite is in an elliptical orbit about a planet.
Its energy is p2/(2m) - GMm/r = pmax2/(2m) - GMm/rmin
= -GMm/(2a),
where M is the mass of the
planet, m is the mass of the satellite, and a is the length of the semi-major axis of the orbit.
(a) At the perigee of the orbit the satellite fires an engine and receives an impulse
of magnitude Δp << pmax perpendicular to the direction of its
velocity. During the very short duration of the impulse, the change in
position of the satellite is negligible.
To first order in Δp, find the change in the length of the semi-major axis
of the satellite's s after the impulse?
(b) At the perigee of the orbit the satellite fires an engine and receives an impulse
of magnitude Δp << pmax in the the direction of its velocity.
To first order in Δp, find the change in the length of the semi-major axis
of the satellite's s after the impulse? By how much does the second
focus of the ellipse shift?
A planet is in a circular orbit around a star of mass M. The radius of
the orbit is R.
(a) What is the orbital speed vo of the planet?
(b) In terms of vo, what would the planet's speed ve
have to be in order for it to escape its star?
(c) An object crosses the planet's orbit with a velocity component vr
= -vo (directed towards the star) and vt = vo (tangential
to the orbit). Describe the object's orbit. Calculate the distance of
closest approach to the star for this object.