A planet orbit?

Question

Suppose there is a planet have a mass of Mp, orbiting a star with mass Ms.
When measured in the same time, the planet have the position vector of (Px i + Py j) and velocity vector of (Vx i + Vy j).
How can we calculate the the time needed for the planet to orbit the star, the orbit eccentricity and the aphelion and perihelion distance of the planet?

The position vector coordinate center is of course the sun, see this URL for more information.

http://www.geocities.com/orichalc_of_moon/Special/GPIntro.html

Answer 1

You need to know angular momentum and energy.

Angular momentum:
L = Mp R (x) V = Mp (Rx, Ry) (x) (Vx, Vy) = Mp [Rx Vy - Ry Vx]

Kinetic energy:
K = 1/2 MpV² = 1/2 Mp (Vx² + Vy²)

Potential energy:
P = -GMsMp/R = -GMsMp / √(Rx² + Ry²)

Total mechanical energy:
E = P + K = -GMsMp / √(Rx² + Ry²) + 1/2 Mp (Vx² + Vy²)



For further calculations we will need these two values:
Lo = = Mp [Rx Vy - Ry Vx]
Eo = Mp [ -GMs / √(Rx² + Ry²) + 1/2 (Vx² + Vy²) ]

Velocity of planet can be decomposed as lateral
velocity Vl plus radial velocity Vr. Kinetic energy
is then given by
K = 1/2 MpV² = 1/2 Mp (Vl² + Vr²).

At the same time conservation of angular momentum
is conveniently rewritten as
L = Mp R x Vl = Lo,
and
Vl = Lo/Mp 1/R.

Kinetic energy can thus be rewritten as
K = 1/2 Mp (Vl² + Vr²). = 1/2 Mp ((Lo/Mp)² 1/R² + Vr²)



Pericenter and apocenter can be found from condition
that radial velocity is zero:
K = 1/2 Mp ((Lo/Mp)² 1/R² + 0²) = 1/2 Lo²/Mp 1/R²

Conservation of energy requires that
Eo = K + P = -GMsMp/R + 1/2 Lo²/Mp 1/R²

*********
Eo R² + GMsMp R - 1/2 Lo²/Mp = 0
*********

The latter is quadratic equation which has two solutions
Rp and Ra for pericenter and apocenter respectively.
Major semiaxis of ellipse is accordingly (Rp + Ra) / 2.

Major semiaxis is all you need to find orbital period
according to Kepler law.

Answer 2

Double planets are achievable and a minimum of one exists. Pluto and Charon are the two in tidal lock with one yet another and that they orbit a uncomplicated midsection of gravity between the worlds. it incredibly is the midsection of gravity that orbits the solar. Double planets could be uncommon(?), yet there is not any reason they are able to't exist.

Answer 3

This is not an easy problem. You may get some help here http://en.wikipedia.org/wiki/Orbital_mechanics