A satellite with a mass of ms = 7.00 × 103 kg is in a planet's equatorial plane in a circular "synchronous" orbit. This means that an observer at the equator will see the satellite being stationary overhead (see figure below). The planet has mass mp = 8.59 × 1025 kg and a day of length T = 1.1 earth days (1 earth day = 24 hours).
(a) How far from the center (in m) of the planet is the satellite?
(b) What is the escape velocity (in km/sec) from this orbit?
To determine the distance of the satellite from the center of the planet in part (a), we can use the formula for the gravitational force between two objects:
F = G * (m1 * m2) / r^2
where
F is the gravitational force,
G is the gravitational constant (approximately 6.67430 × 10^-11 N m^2 / kg^2),
m1 and m2 are the masses of the two objects, and
r is the distance between the centers of the two objects.
In this case, the satellite is in a circular orbit, so the gravitational force between the satellite and the planet provides the necessary centripetal force (Fc) to keep it in orbit:
Fc = (m * v^2) / r
where m is the mass of the satellite, v is the velocity of the satellite, and r is the radius of the orbit (distance from the center of the planet).
Since the satellite is in a synchronous orbit, it means that its orbital period (T) is equal to the planet's rotational period. The velocity of the satellite can be calculated using:
v = (2 * π * r) / T
Since we want to find the radius r, we can rearrange the equation for v:
r = (v * T) / (2 * π)
Substituting this value of r in the equation for the gravitational force (Fc), and equating it to the gravitational force (F) between the satellite and the planet, we can solve for r.
Now let's calculate the distance of the satellite from the center of the planet (r) by following these steps:
Step 1: Calculate the velocity of the satellite (v) using the given period (T) and radius r.
v = (2 * π * r) / T
Step 2: Substitute the given values into the equation and solve for v.
Given:
T = 1.1 earth days = 1.1 * 24 * 60 * 60 seconds
r = ?
Substituting these values:
v = (2 * π * r) / (1.1 * 24 * 60 * 60)
Step 3: Calculate the mass of the satellite (m) using the given value.
Given:
ms = 7.00 × 10^3 kg
Step 4: Calculate the gravitational force (F) between the satellite and the planet using the formula:
F = G * (m1 * m2) / r^2
Given:
G = 6.67430 × 10^-11 N m^2 / kg^2
mp = 8.59 × 10^25 kg
F = ?
Step 5: Equate the gravitational force (F) to the centripetal force (Fc):
Fc = F = (m * v^2) / r
Step 6: Solve the equation to find the radius of the orbit (r).
Now, let's move on to part (b) to determine the escape velocity.
The escape velocity is the minimum velocity required to overcome the gravitational pull of a celestial body and escape its gravitational field. It can be calculated using the formula:
v_escape = √(2 * G * m1 / r)
where
v_escape is the escape velocity,
G is the gravitational constant,
m1 is the mass of the planet, and
r is the distance between the object and the center of the planet.
Substituting the given values into the equation, we can calculate the escape velocity.