The shuttle must perform the de-orbit burn to change its orbit so that the perigee, the point in the orbit closest to Earth, is inside of Earth's atmosphere. De-orbit maneuvers are done to lower the perigee of the orbit to 60 miles (or less). An altitude of 60 miles is important because this is where the orbiting spacecraft is recaptured by Earth’s gravity and re-enters Earth’s atmosphere.

Question

The shuttle must perform the de-orbit burn to change its orbit so that the perigee, the point in the orbit closest to Earth, is inside of Earth's atmosphere. De-orbit maneuvers are done to lower the perigee of the orbit to 60 miles (or less). An altitude of 60 miles is important because this is where the orbiting spacecraft is recaptured by Earth’s gravity and re-enters Earth’s atmosphere.

Calculate the minimum change in velocity (delta v or ∆v) required for the space shuttle to decrease its altitude to 60 miles when it’s orbiting with an apogee of 246 miles and a perigee of 213 miles above the surface of Earth.

Answer 1

To calculate the minimum change in velocity (delta v or ∆v) required for the space shuttle to decrease its altitude to 60 miles, we need to consider the difference between the current perigee and the desired perigee.

Step 1: Find the current perigee altitude above the surface of the Earth.
The perigee is given as 213 miles above the surface of Earth.

Step 2: Find the desired perigee altitude above the surface of the Earth.
The desired perigee altitude is 60 miles above the surface of Earth.

Step 3: Calculate the difference in altitude.
Difference in altitude = Current perigee altitude - Desired perigee altitude
Difference in altitude = 213 miles - 60 miles = 153 miles

Step 4: Convert the altitude difference to meters.
1 mile is approximately equal to 1609.34 meters.
Altitude difference in meters = Difference in altitude * 1609.34 meters
Altitude difference in meters = 153 miles * 1609.34 meters = 246343.02 meters

Step 5: Calculate the change in velocity (∆v) required.
The change in velocity required to decrease altitude is approximately equal to twice the orbital velocity change at the perigee.

∆v = 2 * √(2 * G * M / R_initial - 2 * G * M / (R_initial + Altitude difference))
where,
G = Gravitational constant = 6.67430 * 10^(-11) m^3/(kg * s^2)
M = Mass of Earth = 5.97219 * 10^24 kg
R_initial = Radius of initial orbit = Average radius of the Earth + Current perigee altitude

Step 6: Calculate the change in velocity (∆v).
∆v = 2 * √(2 * G * M / R_initial - 2 * G * M / (R_initial + Altitude difference))
∆v = 2 * √(2 * 6.67430 * 10^(-11) * 5.97219 * 10^24 / (6,371,000 + 246,343.02) - 2 * 6.67430 * 10^(-11) * 5.97219 * 10^24 / 6,371,000)
(steps 4 and 5)

Now, you can use a calculator or any mathematical software to evaluate this expression and get the ∆v required for the space shuttle to decrease its altitude to 60 miles.