A satellite of mass m is in Keplerian orbit around a parent body of mass M >>m. The satellite has total energy E and angular momentum l, where angular momentum is measured about the parent body and the zero of potential energy is at infinity. Detnermine the semimajor axis a and the eccentricity e of the orbit, in therms of the above quantities and Newton'ss constant G (use the symbol, not the numerical value). HINT: Find the relation amonb the energy, angular momentum and orbit raduis at the apside of the orbit. Solve this relation for the apsidal radii r, and use thed fact that the apsidal radii are r= a(1+-e)

Question

A satellite of mass m is in Keplerian orbit around a parent body of mass M >>m. The satellite has total energy E and angular momentum l, where angular momentum is measured about the parent body and the zero of potential energy is at infinity. Detnermine the semimajor axis a and the eccentricity e of the orbit, in therms of the above quantities and Newton'ss constant G (use the symbol, not the numerical value). HINT: Find the relation amonb the energy, angular momentum and orbit raduis at the apside of the orbit. Solve this relation for the apsidal radii r, and use thed fact that the apsidal radii are r= a(1+-e)

Answer 1

To determine the semimajor axis (a) and eccentricity (e) of the orbit, we need to use the relation between energy, angular momentum, and the radius of the apside of the orbit.

The energy of the satellite in its orbit is given by:

E = -G * M * m / (2 * a)

where G is Newton's constant, M is the mass of the parent body, m is the mass of the satellite, and a is the semimajor axis of the orbit.

The angular momentum of the satellite is given by:

l = m * sqrt(G * M * a * (1 - e^2))

where l is the angular momentum and e is the eccentricity of the orbit.

Now, we need to find the relation between the energy, angular momentum, and the apsidal radius (r*) of the orbit.

The apsidal radii are r* = a * (1 ± e), which means at the apsis of the orbit (either pericenter or apocenter), the distance of the satellite from the parent body is a(1 + e) or a(1 - e), respectively.

At the apside of the orbit, the satellite's energy is equal to its potential energy, which is given by:

E = -G * M * m / (2 * r*)

Substituting the relation for apsidal radii:

E = -G * M * m / (2 * a(1 ± e))

Now, we can solve this relation for the apsidal radii (r*):

r* = a(1 ± e) = -G * M * m / (2E)

Now, we need to substitute the expression for angular momentum (l) in terms of a and e:

m * sqrt(G * M * a * (1 - e^2)) = -G * M * m / (2E)

Squaring both sides and canceling out the mass (m) on both sides, we get:

G^2 * M^2 * a^2 * (1 - e^2) = G^2 * M^2 * m^2 / (4E^2)

Cancelling out the common terms, we get:

a^2 * (1 - e^2) = m^2 / (4E^2)

Now, we can solve this equation for the semimajor axis (a) and eccentricity (e):

a = sqrt(m^2 / (4E^2 * (1 - e^2)))

e^2 = 1 - (m^2 / (4E^2 * a^2))

Therefore, the semimajor axis (a) and eccentricity (e) of the orbit, in terms of the given quantities and Newton's constant (G), are:

a = sqrt(m^2 / (4E^2 * (1 - e^2)))

e^2 = 1 - (m^2 / (4E^2 * a^2))