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Examples of orbits

Figure 1: Geostationary orbit and polar orbit.

Three types of orbits are of particular interest: low Earth orbits (LEO), which goes up to about 2000 km; medium Earth orbits (MEO), which are between LEO and GEO (see below); and geostationary Earth orbits (GEO), which are at 42 164 km. Hardly any satellites orbit Earth above GEO (with one notable exception, which will be discussed later).

Most satellites in LEO orbit above 300 km, where the reduced atmospheric drag make it possible to maintain a reasonably stable altitude without spending too much fuel on orbital maintenance. Nonetheless, periodic burns are needed. The International Space Station (ISS) orbits at 350–400 km, and its altitude is chosen as a trade-off between the cost of reaching orbit and the fuel needed to be transported for altitude maintenance. 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, allowing satellites to stay in orbit for years without ever needing to compensate for the drag.

A geostationary satelitte orbit the Earth once every sidereal day (about 23 hours and 56 minutes, since the Earth moves around the Sun while rotating around its own axis). This means that the satellite, as viewed from Earth, stays still above a fixed location on the ground. This make it much easier to receive TV signals and communication from it, as practically no tracking is 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.

hohmann_2

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

See how astronauts reach the ISS in this video:

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

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

For an apogee at r = 42 \; 164\; \text{km}, we find the velocity

v = \sqrt{2\cdot 398 \; 600 \; \text{km}^3/\text{s}^2 \left( \dfrac{1}{42\; 164 \; \text{km}} - \dfrac{1}{2\cdot 24 \; 582 \; \text{km}} \right)} = 1.64 \ \text{km}/\text{s}.

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

v = \sqrt{\dfrac{\mu}{r}} = 3.08\ \text{km}/\text{s}.

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

\Delta V_{\text{GTO} \rightarrow \text{GEO}} = 3.08\; \text{km}/\text{s} -1.64 \; \text{km}/\text{s} = 1.44 \ \text{km}/\text{s}.

For a satellite with an initial wet mass (including fuel) of 4 000 kg and a motor with I_\text{SP} = 3\, 000 \; \text{m}/\text{s}, the final fuel after burn is

M_\text{f} = M_\text{i} \exp \left( - \dfrac{\Delta V}{I_\text{SP}} \right) = 4 \, 000 \, \exp \left( - \dfrac{1.44 \; \text{km}/\text{s}}{3 \; \text{km}/\text{s}}} \right) = 2\, 475 \ \text{kg},

making the propellant needed for the insertion m_\text{p} = 4 \, 000\; \text{kg} - 2\, 475\; \text{kg} = 1 \, 525 \; \text{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!