This tutorial will help you get started in simple Astronautics calculations using the Systems Tool Kit from AGI or Python. We will use the International Space Station as our orbiting body around the Earth.
The ISS is the largest man-made satellite ever made. It’s been orbiting the earth since 1998 and estimates say it has cost about $100 billion so far. It can be seen with the naked eye from ground. Since the year 2000 it has been continuously occupied by different astronauts from different nations. To read more about it go to: http://en.wikipedia.org/wiki/International_Space_Station
This tutorial will help you calculate some interesting orbital properties of the ISS like it’s orbital speed and period. I’ll give you some of the relevant variables and equations so you can easily compute everything. The main purpose of this tutorial is to get you started with powerful software tools that can help you solve real problems.
With a simple calculator go trough the given equations and get the relevant numbers. Just because I give you the numbers you should’t trust them blindly. Then you will verify your hand made numbers using STK or Python.
Don’t worry if you don’t understand all the terms yet. Later you will learn more about these terms like the true anomaly and orbital energy (read this and this to get started right away) . These steps were made so you can easily compute the orbital speed and period of the ISS.
Here is the data and the relevant equations (make sure you use the right units):
Earth Radius $R_e = 6371 km$
Earth gravitational parameter $\mu = 3.986×10^5 km^3/s^2$
Perigee = 418 km
Apogee = 426 km
$R_p = 418 + Re$
$R_a = 426 + Re$
1) Find the eccentricity of the orbit (e)
$e = \frac{R_a – R_p}{R_a + R_p} = 0.000588841454438$
it’s very small!!! as expected, meaning the orbit is almost circular
2) Find the semi-major axis (a)
$2a = Ra + Rp => a = 6793.0 km$
3) Find the specific mechanical energy for the ISS orbit (espilon)
$\epsilon = – \frac{\mu}{2a} = -29.339025467392904 km^2/s^2$
4) Find the altitude (R) of the ISS when the true anomaly is 90deg (usually represented by the greek letter “nu”)
$R = \frac{a(1-e^2)}{1+e \times cos(\nu)} = 6792.9976446341825 km$
don’t forget to convert deg to radians when you use this equation.
5) Find the orbital speed (at nu = 90deg)
$V = \sqrt{2(\frac{\mu}{R} + \epsilon)} = 7.660163 km/s$
6) Find the orbiting period of the ISS
$P = 2\pi\sqrt{\frac{a^3}{\mu}} = 5571.90 sec = 92.87 min = 1.55 hr$
because the results are given in seconds I convert to minutes and hours to get a better feeling of it how long it really takes for the ISS to orbit the Earth once.
BTW, here’s a really cool video of the Earth taken from the ISS at 7.6 km/s
Now follow these links to check if the above equations are correct:
Systems Tool Kit (STK) intro
go to http://www.spacemig.com/stk-intro/
