# [amsat-bb] Re: Path to HEO

Patrick Strasser patrick at wirklich.priv.at
Mon Apr 29 12:50:19 PDT 2013

```schrieb Stefan Wagener on 2013-04-29 01:29:

> LEO (low earth orbit) from 400 to 2000km or something like that
> HEO (high earth orbit) >20000km
>
> Going from 310 to 700km in a year is not doing us anything. We need a
> highly elliptical orbit (Apogee ~60000km, Perigee 900km) similar to AO-40
> to allow for cross continental communication.

Let's check with some maths (*):

Energy at 300 km:
-----------------

Total Energy = kinetic energy + potential energy
E_kin     + E_pot =
(m*v^2)/2 + m*g*h

For simplicity, we choose
mass as 1 kg,
h = 300km,
v = 1st cosmic velocity =~ 7100m/s

1*7200² + 1*9.81*300x10^3 = 2.8148x10^7 [Joule]

Potential Energy is some 10% of the total energy.
As this is for one kg of mass, and m goes linear in the above equations,
you can scale with the mass of your satellite.

Energy at 36000 km:
-------------------

Speed from radius and time for one orbit (1 day=84600 sec)
v = 2*r*pi/t = 2*36x10^6*3.14159265/84600 = 2673.7 m/s

E_kin + E_pot =
(m*v^2)/2 + m*g*h =
3.57x10^6 + 3.53^8 =~ 3.56×10^8 [Joule]

Now kinetic energy is only about 1% of the total energy!

A LEO has about 8% the energy of an GEO. The satellite needs 3.29x10^8
J/Kg Energy to get from LEO to GEO.

Lets say it's 10kg and has 50 W of power for thrust.
3.3x10^8 * 10 = 3.3x10^9 J thrust

1 Joule is 1 Watt / 1 second, 1 Watt second = 1 Joule
1 Watt day = 84600 Joule = 8.46x10^4 Joule

Our 50 Watt ion drive can increase the energy by 4.23x10^6 Joule a day.
How much days will LEO to GEO take:

We have some 10^8 divided by some 10^6, it's a matter of months, the
calculation says 77.8. But we should be satisfied to get an order of
magnitude after the rough assumtions and estimations made before.
If I made a mistake above, maybe this is off by an factor of 10, then
it's 2 year. Still fine!

Of course you have to count in the gas you want to ionize, which reduces
the weight over time (but I was really bad at differential equations and
would not get that right), and maybe the weight and power estimations
are not very realistic, and using steady thrust instead of short
impulses decreases efficiency in orbit changes, and changing from polar
to equatorial orbit takes extra energy, and maybe an elliptic orbit
takes less energy, and maybe some inaccuracies more. But this does not
matter:

In the end, it seems that changing from LEO to GEO or HEO is possible in
sensible time.

Regards

Patrick

(*) Disclaimer: This is High School maths, please double check and
correct my calculations
--
Engineers motto: cheap, good, fast: choose any two
Patrick Strasser <patrick at wirklich priv at>
```