You are an astronaut in the space shuttle pursuing a satellite in need of repair. You are in a circular orbit of the same radius as the satellite (450 km above the Earth), but 24 km behind it.

How long will it take to overtake the satellite if you reduce your orbital radius by 1.4 km?
By how much must you reduce your orbital radius to catch up in 7.0 h?

Assume, for discussion purposes, a satellite and the shuttle are in the same 91 minute circular orbit, the satellite 2 minutes ahead of the space shuttle.

The shuttle fires its on-orbit thrusters to reduce its orbital velocity. In doing so, the space shuttle drops into an elliptical orbit with a period 1 minute slower than the original circular orbit.

In 90 minutes, the space shuttle returns to the firing point of the elliptical orbit, now only 1 minute away from the satellite.

Remaining in the same elliptical orbit once more, the space shuttle ultimately catches up with the satellite. It now fires its thrusters to speed up the space shuttle to the same circular orbit velocity that it had when the catch up operation started.

As for your specific problem:
Assume an earth radius of 6378km
Orbital altitude = 450km
Orbital radius = radius is 6378 + 450 = 6828km or 6,828,000meters.
Earth’s gravitational constant GM = µ = 3.986365x10^14m^3/sec^2
The orbital velocity is Vc = sqrt(µ/r) = 7641m/s.
The orbital period T = 2(Pi)sqrt(a^3/µ) = 5615sec. = 93.58minutes.
…(a = semimajor axis = r for the circular orbit)
Reducing the velocity of the space places the space shuttle into an elliptical orbit with an apogee of 6826km and a new perigee of 6828 – 1.4 = 6826.6 km and a semi-major axis of 6827.3km.
The period of this new orbit becomes 5614sec. Placing the shuttle 1 second closer to the satellite.
The circular orbit perimeter is 6828(2)3.14 = 42,901.6km.
The satellite is 24(5615)/42,901.6 = 3.14sec. ahead of the space shuttle.
Therefore, after 3 of the new elliptical orbits are completed, the two vehicles are within .14sec. of one another which can be effectively accomplished with the reaction control system.

Addendum to my earlier response.

A reduction in velocity of only 1.5m/sec. will drop the space shuttle into an ever so slightly elliptical orbit with an apogee of 6828km, a perigee of 6825.6km. and a period of 5611.86 sec. or 93.53min., allowing the shuttle to catch up with the satellite in one orbit.

A 7 hour closing time is somewhat impractical.

Well, it seems like you're in quite a "space chase"! Let's calculate the time it will take for you to overtake the satellite and figure out how much reduction in your orbital radius you need to catch up.

To calculate the time it takes to overtake the satellite when reducing your orbital radius by 1.4 km, we can start by finding the initial orbital distance between you and the satellite. Since you are 24 km behind the satellite, the total distance between you and the satellite is 450 km (radius of circular orbit) + 24 km = 474 km.

Now, let's find the new orbital radius after reducing it by 1.4 km. The new orbital radius would be 450 km - 1.4 km = 448.6 km.

The difference in distance between you and the satellite is the initial distance of 474 km minus the new orbital radius of 448.6 km, which is 25.4 km.

To calculate the time it takes to close this gap, we can divide the distance by the relative speed. In this case, the relative speed is the speed of the satellite minus your speed. Let's assume your speed is the same as the satellite's speed, so the relative speed would be 0.

When dividing the distance of 25.4 km by the relative speed of 0 km/h, we end up with a time of infinity. So unfortunately, reducing your orbital radius by 1.4 km won't help you overtake the satellite in this case.

Now let's figure out how much you need to reduce your orbital radius to catch up within 7.0 hours. To do this, we can use the formula: reduction in orbital radius = (relative speed) x (time).

Since relative speed is 0, multiplying it by any time would still give us 0. Therefore, you don't need to reduce your orbital radius at all to catch up in 7.0 hours. You can simply maintain your current orbit and coordinate with Mission Control for any necessary maneuvers.

Remember, in space, it's not all about speed but also about the right strategy. So take your time, float around, and don't forget to enjoy the cosmic view!

To find the time it takes to overtake the satellite when you reduce your orbital radius by 1.4 km, we need to calculate the time it takes to complete one orbit at both the original and reduced radii and then determine the difference.

Step 1: Calculate the orbital period at the original radius.
The orbital period of a satellite in circular orbit can be calculated using the formula: T = 2π√(r³/GM)
Where:
T = orbital period
r = orbital radius
G = gravitational constant
M = mass of the Earth

At the original radius of 450 km above the Earth, the orbital period can be calculated as follows:
T1 = 2π√((450+6371)³/(6.67430×10^(-11)×5.97219×10^24))

Using a scientific calculator or software, calculate T1.

Step 2: Calculate the orbital period at the reduced radius.
To calculate the new orbital period after reducing the radius by 1.4 km, make use of the formula from Step 1, but substitute the reduced radius.

T2 = 2π√((448.6+6371)³/(6.67430×10^(-11)×5.97219×10^24))

Using a scientific calculator or software, calculate T2.

Step 3: Determine the time it takes to overtake the satellite.
The time it takes to overtake the satellite can be calculated by subtracting the orbital period at the reduced radius from the orbital period at the original radius.

Time taken to overtake = T1 - T2

Calculate the difference using the values obtained in Steps 1 and 2.

To find out how much the orbital radius must be reduced to catch up in 7.0 hours, we can use a similar process.

Step 1: Calculate the orbital period at the original radius as explained in Step 1 above.

Step 2: Calculate the time it takes for the satellite to complete one orbit.
Using the formula: T = 2π√(r³/GM), solve for T using the original radius.

Calculate T.

Step 3: Subtract 7.0 hours from the orbital period.
Subtract 7.0 hours from T to find the difference.

Step 4: Determine the change in orbital radius that causes this difference in time.
Find the change in orbital radius using the formula: Δr = (GM/4π²) x (ΔT/T²)
Where:
Δr = change in orbital radius
G = gravitational constant
M = mass of the Earth
ΔT = difference in time
T = original orbital period

Calculate the change in orbital radius using the values obtained in Steps 2 and 3.

Note: It is important to use accurate values for the gravitational constant (G), the mass of the Earth (M), and to ensure consistency in units (such as kilometers and meters) when performing the calculations.

If you reduce your orbital RADIUS, you will acquire a different small circular orbit and will never get to the satellite. You will pass between it and the earth.

This is a poorly conceived question, in my opinion.

What the astronaut needs to do is apply reverse thrust to acquire an elliptical orbit. Its semimajor diameter will be reduced, and it will return to meet the satellite on the next orbit.