Examples of orbits

Figure 1: Geostationary orbit and polar orbit.

Three types of orbits are of most interest: low Earth orbit (LEO) up to about 2000 km, medium Earth orbit (MEO) from 2000 km and to geostationary orbit (GEO) (at 42 164km), and then GEO. Few, if any, satellites above that are orbiting around Earth (with one exception, which we will see later).

Satellites in LEO are usually found above 300 km, where the satellites can maintain a stable attitude for a long time due to decreasing atmospheric drag, though they are dependent on burning their motors to maintain their altitude. At the International Space Station (ISS) this is done routinely to maintain its altitude of 350-400 km. The altitude is a trade-off between the costs of reaching the orbit and bringing fuel to the space station to maintain its altitude.  The ISS has an inclination of about 52 degrees, making it easier to reach from the USA and Russia which frequently send cargo and astronauts. Above 500-600 km the atmospheric drag is negligible. The satellites may stay in orbit for years without compensating for drag.

Geostationary orbit where the satellites orbit Earth once every sidereal day (about 23 hours and 56 minutes, to compensate for an Earth year not being exactly 365 days). This means that the satellite is exactly over the same point over the ground, as if the satellite was stationary on the sky. This make it much easier to receive TV signals and communication from it, as no tracking of the geostationary satellites are needed.

Another important orbit is at about 20 000 km of altitude (orbital period of approx. 12 hours) where most global navigation satellite system constellations are found.


Figure 2: A satellite can start in the lower (blue) orbit and reach the final green orbit by going into a Hohmann orbit, here seen in red. The satellite fires its engine twice to do this, once at the perigee of the Hohmann ellipse and once at the appogee.

The last satellite orbit we will discuss is not an orbit that satellites operate in, but merely a way of transportation to its final orbit. This orbit is called a Hohmann orbit. Imagine a satellite launched into a circular orbit of 350 km altitude. The satellite is intended to reach a circular orbit with an altitude of 5000 km. The satellite at 350 km altitude increases its orbit with a change in velocity of \Delta V_1, and this changes the orbit to an elliptical orbit with perigee at 350 km and apogee at 5000 km since \Delta V_1 is carefully calculated in advance. After half an orbital period the satellite reaches the apogee, the it does a second burn with \Delta V_2 to change the orbit once again to a circular orbit. See Figure 2. Note that Hohmann orbits can be used to increase and decrease the orbits altitude, and that the initial and final orbit needs not to be circular. Hohmann orbits are always elliptical and requires two changes in velocity (two motor burns). For orbital changes where the ratio between the lower and higher semi-major axis is not very large Hohmann transfers are the most efficient. In some rear cases, bi-elliptical transfers can be more efficient.

In practice satellites intended to go in higher orbits than low LEO orbits are usually launched straight into a Hohmann orbits. One example is the geostationary transfer orbits (GTO), which carry communication satellites to GEO. Launching straight into GEO makes the satellite only needing to burn its own motors when the apogee in the GTO is reached (called orbital insertion).

See how astronauts reach the ISS in this video:

GTO has a semi-major axis of a = 24 582 \, \mathrm{km} Using the energy conservation formula v^2 - 2 \mu/ r = - \mu/a we can write

v = \sqrt{2\mu \left( \frac{1}{r} - \frac{1}{2a} \right)}.

At apogee at r = 42 \, 164\,\mathrm{km} we can find the velocity

v = \sqrt{2\cdot 398 \, 600 \frac{\mathrm{km}^3}{\mathrm{s}^2} \left( \frac{1}{42\, 164 \, \mathrm{km}} - \frac{1}{2\cdot 24 \, 582 \, \mathrm{km}} \right)} = 1.64 \, \frac{\mathrm{km}}{\mathrm{s}}.

The orbital velocity at GEO (at the same altitude) is

v = \sqrt{\frac{\mu}{r}} = 3.08 \frac{\mathrm{km}}{\mathrm{s}}.

The change in velocity, \Delta V, needed for orbital injection to GEO is therefore

\Delta V_{GTO \rightarrow GEO} = 3.08-1.64 \, \frac{\mathrm{km}}{\mathrm{s}} = 1.44 \, \frac{\mathrm{km}}{\mathrm{s}}.

For a satellite with an initial wet mass (including fuel) of 4 000 kg and a motor with Isp = 3 000 m/s, the final fuel after burn is

M_f = M_i \exp \left( - \frac{\Delta V}{I_{SP}} \right) = 4 \, 000 \, \exp \left( - \frac{1.44 \, \mathrm{km/s}}{3 \, \mathrm{km/s}}} \right) = 2\, 475 \, \mathrm{kg},

making the propellant needed for the insertion m_p = 4 \, 000 - 2\, 475 = 1 \, 525 \, \mathrm{kg}.

After orbital injection, the satellite needs some propellants for station keeping, burning the motors to compensate for secondary effects. Some of these effects is accounting for Earth not being perfectly spherical, sun pressure and gravitational forces from other celestial objects. After the end of life of a geostationary satellite, being that GEO is such an important object for the modern society, the satellites is moved to a higher orbit called a graveyard orbits only used for decommissioned satellites.

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This article is a part of a pre-course program, used by NAROM in different courses, for example Fly a Rocket!