An asteroid is discovered to have a tiny moon that orbits it in a circular path at a distance of 122 km and with a period of 43 h. The asteroid is roughly spherical (unusual for such a small body) with a radius of 19 km.

(a) Find the acceleration of gravity at the surface of the asteroid.
and
(b) Find the escape velocity from the asteroid.
I can find the escape velocity once I calculate part a and for part a I know the formula, but how do I find M to plug in?

To find the acceleration of gravity at the surface of the asteroid (part a), you will need to calculate the mass of the asteroid. The formula you mentioned requires the object's mass (M) and the radius (r) to calculate the acceleration of gravity (g) using the equation:

g = G * (M / r^2)

where G is the universal gravitational constant (approximately 6.674 × 10^-11 Nm^2/kg^2).

To find the mass (M) of the asteroid, you can use the formula for centripetal force, which is given by:

F = (M * v^2) / r

where F is the gravitational force acting on the moon, v is the orbital velocity of the moon, and r is the distance between the moon and the center of the asteroid.

First, you need to find the orbital velocity (v) of the moon using the period (T) of the orbit. The formula for orbital velocity is given by:

v = (2π * r) / T

where r is the distance between the moon and the center of the asteroid, and T is the period of the orbit.

Once you have the orbital velocity (v), you can substitute it into the centripetal force equation to solve for the mass (M). Rearranging the equation, we get:

M = (F * r) / v^2

Now, calculate the gravitational force (F) using the formula:

F = G * (m * M) / r^2

where m is the mass of the moon. Since the gravitational force acting on the moon is equal to the centripetal force, we can integrate these equations to solve for M.

With the calculated mass (M), you can then plug it into the formula for the acceleration of gravity (g) to find the answer to part a.

To find the escape velocity from the asteroid (part b), you can use the formula:

v_escape = sqrt((2 * G * M) / r)

where G is the universal gravitational constant, M is the mass of the asteroid, and r is the radius of the asteroid. Substituting the mass value calculated in part a, you can find the escape velocity.

To find the acceleration of gravity at the surface of the asteroid, you need to calculate the mass of the asteroid, denoted by M. Here's how you can find it:

First, let's find the period of the tiny moon's orbit using Kepler's Third Law. The formula for the period of a circular orbit is:

T^2 = (4π^2 R^3) / (G M)

where T is the period, R is the distance between the asteroid and its moon, G is the universal gravitational constant, and M is the mass of the asteroid. Rearranging the formula, we get:

M = (4π^2 R^3) / (G T^2)

Now, substitute the known values into the equation:
R = 122 km = 122,000 m (converted to meters)
T = 43 hours = 43 x 60 x 60 seconds (converted to seconds)
G = 6.67430 x 10^-11 m^3 kg^-1 s^-2 (universal gravitational constant)

M = (4π^2 * (122,000)^3) / (6.67430 x 10^-11 * (43 x 60 x 60)^2)

Solve this equation to find the value of M.

Once you have the mass of the asteroid, you can proceed to find the escape velocity using the formula:

v = sqrt(2GM / R)

Where v represents the escape velocity, G is the universal gravitational constant, M is the mass of the asteroid, and R is the radius of the asteroid.

Substitute the known values into the equation and calculate the escape velocity.

Note: It's important to remember that the mass of the tiny moon is negligible compared to the mass of the asteroid, so you can safely assume that the mass of the asteroid is equal to the combined mass of the asteroid and its moon.