Intro: Calculate Satellite Orbital Time and Speed
I am a member of my school's cubesat team, where we basically design and build a 10 cm cubed satellite. I became involved with orbital software, but this was somewhat faulty and we needed a way to confirm our results. Naturally, I dug deeper, and found out that in fact we could easily calculate the satellite speed and orbit time, without knowing its mass or anything else. All we needed was the altitude.
This instructable will detail the process to find orbit time and speed if you know your orbit altitude. In order for a satellite to be in orbit, it's centripetal force and the force of gravity have to be equal, otherwise it would enter the atmosphere and burn or be flung far out into the cosmos.
Looking at the situation, you would think that mass would be involved, but because of the mathematics, the mass of the satellite is canceled out. You don't need to know the satellite mass or any other data- the only real outside force is gravity in the situation.
This only works for a circular orbit. With an elliptical orbit, math becomes more complex and requires more than just simple algebra and plugging in numbers.
Step 1: Find Your Radius of Orbit/ Altitude
You do not need to know anything else. You don't need satellite mass, drag, or other things. Mass cancels out for these calculations, and drag doesn't apply to our theoretical calculations. Drag is very small in any orbits beyond Low Earth Orbit (LEO) and even in LEO drag has a small effect the majority of the time.
To find the radius of the orbit, you would need to get the altitude and the radius of the earth and add them together. I did this in meters so I could get meters per second orbital speed out of the formula.
Step 2: Plug Into Equation
Plug all your variables into the equation. G is the universal gravitational constant, me is mass of earth in kg, and r is your radius of orbit (radius of earth+altitude).
The universal gravitational constant is 6.67408 × 10-11 m3 kg-1 s-2
the mass of the earth in kg is 5.972 × 10^24 kg
and the radius is what you should have already calculated. This should be in meters.
The output of the equation will be your orbital speed in meters/ second. You can convert this easily to km/h or mph or any other speed unit you would like.
Step 3: Calculate Orbit Time
Now that you have the orbit speed, finding the orbit time is a breeze.
by definition, speed = distance / time
We can rearrange this so that time= distance / speed.
Once we have this, we can calculate the distance of the orbit by using the circumference equation
C= 2* pi * r
C is circumference, pi is the mathematical number, and r is your radius of orbit (include radius of earth)
For the distance units of r, they should match that of your speed so if you had r in meters your speed should be in meters/ sec, and so on.
We then take the distance, and divide that by speed. The units of the time are whatever the units of your speed are, so if you did meters per second your time unit would be seconds.
Step 4: Taking It Further
Now that we have the time per orbit, we can easily take this to the next level and calculate orbits per day. Just take the total amount of time per day, and divide it by your time per orbit.
So if there are 24 hours in a day, and I know one orbit takes 3 hours, I would divide and get 8 orbits per day. This works for any unit of time as long as they are matching.
Beyond this, you can apply this knowledge to satellites, physics problems, and a variety of other situations.
The international space station orbits at about 408 km- take this and find it's speed! Once again, this only works for circular orbits, but many satellites such as GPS satellites or the ISS are circular.
Even with the limitations of a circular orbit, I found it fascinating to learn how to calculate the orbit speed of any given circular orbiting satellite. It's crazy to me that I can find the orbit speed within 5 minutes of the international space station, or even a weather or GPS satellite.
Step 5: Excel Calculator
With the prior knowledge, it was fairly easy for me to create a calculator in Microsoft Excel. This allows you to input your altitude, and it punches out the numbers for you.
Step 6: Thanks
Thanks for reading! Feel free to comment if you have any questions.