Central Force and Planet Motion

[Gogsey]

4. For a wide range of orbital radii, stars in the Galaxy (including our sun) move in nearly circular orbits with a speed v0 »220 km/s which is independent of the orbit's radius. Find the force law and the form of the potential energy function for a star moving through the Galaxy (it is not an inverse-square force because the mass of the Galaxy is not all concentrated at the centre; assume it is a central force). Then find an expression for the period of small radial oscillations; apply it to the orbit of the sun (r » 8 kpc) to get a value in years, and compare with the orbital period of the sun around the Galaxy. (A kiloparsec is about $3.086×1016 km.)

5. A satellite of mass m is initially moving at speed v0 in a circular orbit of radius r0 about a planet. Find, in terms of m, v0, and r0, the initial angular momentum, kinetic energy, potential energy, and total energy of the satellite (first, you may need to find the mass of the planet in terms of these quantities.) Then a rocket motor is fired briefly, changing the velocity vector by an amount Dv of magnitude |Dv| = v0/2. Find the angular momentum,
total energy, semimajor axis, and minimum and maximum distance from the planet's centre for the new orbit if Dv is:
a) opposite to v0, or b) radially inward. Give answers in terms of m, r0, and v0. (Hint: For part b), compare the energy and angular momentum at pericentre to the values immediately after the rocket burn, in terms of two unknowns: the speed and radius at pericentre.

Ok so I'm lost for the first question. I have no idea what iot wants me to do.

As for the second question I'm still working on it.

 

[lanedance]

Hi Gogsey

for the very first part i would ask what mass distribution m(r) gives a potential V(r) that results in a constant orbital velocity (equivalently what is the gravitational acceleration g(r), that gives constnat orbital velocity)