Solve Equatorial Orbit: 12 Hours, Radius & Speed

In summary, a satellite is being put into an equatorial orbit with an orbital period of 12 hours. The radius can be found using the equation r= ((6.67×10^(-11) N∙m^2∕kg^2 × 5.98×10^24 kg × (12×60×60 s)^2)/(4pi^2 ))^(1/3) and the orbital speed can be found using V= Radical (GM/r), where G is the gravitational constant, M is the mass of Earth, and r is the radius. To find how many times a day the satellite will be over the same point on the equator, convert the orbital speed into angular velocity and use an

Homework Statement

A satellite is to be put into an equatorial orbit with an orbital period of 12 hours.
Given: 12 Hours = 12 X 60 X 60 seconds

What is the orbital speed?
How many times a day will the satellite be over the same point on the equator if the satellite orbits in the same direction of the Earth's rotation? If it orbits in the opposite direction?

Homework Equations

r= (GMT^2)/(4pi^2 )^1/3
G = Gravitational Constant = 6.67 X 10^-11 N m^2/kg^2
M = Mass of Earth 5.98 X 10^24 kg
T = time

The Attempt at a Solution

Well I started with the radius equation and plugged everything in

r= ((6.67×10^(-11) N∙m^2∕kg^2 × 5.98×10^24 kg × (12×60×60 s)^2)/(4pi^2 ))^(1/3)

However I had trouble working it out. Then the other problems just went over my head.

Well you already got the first question, just multiply the numbers.

You already stated that the orbital speed is $$\sqrt{\frac{GM_{earth}}{r}}$$

You know radius already and the other values are constants so find the orbital speed.

Convert orbital speed into angular velocity and do the same for a point on the equator. Then come up with an equation for the angular displacement of the satellite and the point on the equator (angular velocity * time). Then graph these equations and find their intercepts on the domain 0 < t < 86400 (seconds in a day)(note that if the angular displacement differs by a integer multiple of 360 they are technically in the same place). Do something similar for the other scenario.

1. What is an equatorial orbit?

An equatorial orbit is a type of orbit around a planet or other celestial body in which the orbit is aligned with the equator of the body. This means that the orbit is parallel to the planet's rotational axis.

2. Why is the 12-hour period significant in an equatorial orbit?

A 12-hour period is significant because it is the shortest possible orbital period for a satellite in an equatorial orbit. This means that the satellite will complete one full orbit around the planet in 12 hours.

3. How is the radius of an equatorial orbit determined?

The radius of an equatorial orbit is determined by the distance from the center of the planet to the satellite in orbit. This distance is typically measured in kilometers or miles.

4. What is the speed of a satellite in an equatorial orbit?

The speed of a satellite in an equatorial orbit is determined by the orbital period and the radius of the orbit. The speed is calculated using the formula v = 2πr/T, where v is the speed, r is the radius, and T is the orbital period.

5. Can a satellite in an equatorial orbit change its speed?

Yes, a satellite in an equatorial orbit can change its speed. This can be done through the use of thrusters or by adjusting the angle and position of the satellite in its orbit. However, any changes in speed must be carefully calculated in order to maintain the satellite's equatorial orbit.