Frozen orbit
Perturbations can result from natural drifting due to the central body's shape, or other factors.
Typically, the altitude of a satellite in a frozen orbit remains constant at the same point in each revolution over a long period of time.
[1] Variations in the inclination, position of the apsis of the orbit, and eccentricity have been minimized by choosing initial values so that their perturbations cancel out.
[2] This results in a long-term stable orbit that minimizes the use of station-keeping propellant.
For spacecraft in orbit around the Earth, changes to orbital parameters are caused by the oblateness of the Earth, gravitational attraction from the Sun and Moon, solar radiation pressure and air drag.
For a geostationary spacecraft, correction maneuvers on the order of 40–50 m/s (89–112 mph) per year are required to counteract the gravitational forces from the Sun and Moon which move the orbital plane away from the equatorial plane of the Earth.
[citation needed] For Sun-synchronous spacecraft, intentional shifting of the orbit plane (called "precession") can be used for the benefit of the mission.
As a result, the spacecraft will pass over points on the Earth that have the same time of day during every orbit.
Alternatively, if the Sun lies in the orbital plane, the vehicle will always pass over places where it is midday on the north-bound leg, and places where it is midnight on the south-bound leg (or vice versa).
Such orbits are desirable for many Earth observation missions such as weather, imagery, and mapping.
NASA described this in 2006: Lunar mascons make most low lunar orbits unstable ... As a satellite passes 50 or 60 miles overhead, the mascons pull it forward, back, left, right, or down, the exact direction and magnitude of the tugging depends on the satellite's trajectory.
Absent any periodic boosts from onboard rockets to correct the orbit, most satellites released into low lunar orbits (under about 60 miles or 100 km) will eventually crash into the Moon.
Work published in 2005 showed a class of elliptical inclined lunar orbits resistant to this and are thus also frozen.
[7] The classical theory of frozen orbits is essentially based on the analytical perturbation analysis for artificial satellites of Dirk Brouwer made under contract with NASA and published in 1959.
term (corresponding to the fact that the earth is slightly pear shaped), one gets which can be expressed in terms of orbital elements as In the same article the secular perturbation of the components of the eccentricity vector caused by the
Similarly the quadratic terms of the eccentricity vector components in (8) can be ignored for almost circular orbits, i.e. (8) can be approximated with Adding the
to (7) one gets Now the difference equation shows that the eccentricity vector will describe a circle centered at the point
The latter figure means that the eccentricity vector will have described a full circle in 1569 orbits.
[9] For this the analytical expression (7) is used to iteratively update the initial (mean) eccentricity vector to obtain that the (mean) eccentricity vector several orbits later computed by the precise numerical propagation takes precisely the same value.
forces for the numerical propagation one gets exactly the same optimal average eccentricity vector as with the "classical theory", i.e.
When we also include the forces due to the higher zonal terms the optimal value changes to
Assuming in addition a reasonable solar pressure (a "cross-sectional-area" of 0.05 m2/kg, the direction to the sun in the direction towards the ascending node) the optimal value for the average eccentricity vector becomes
This algorithm is implemented in the orbit control software used for the Earth observation satellites ERS-1, ERS-2 and Envisat The main perturbing force to be counteracted in order to have a frozen orbit is the "
With the "modern theory" this explicit closed form expression is not directly used but it is certainly still worthwhile[for whom?]
The derivation of this expression can be done as follows: The potential from a zonal term is rotational symmetric around the polar axis of the Earth and corresponding force is entirely in a longitudinal plane with one component
In the article Geopotential model it is shown that these force components caused by the
term are To be able to apply relations derived in the article Orbital perturbation analysis (spacecraft) the force component
The components of the unit vectors making up the local coordinate system (of which
(between the green points of figure 2) and from equation (12) of the article Geopotential model one therefore obtains Secondly the projection of direction north,
orthogonal to the radial direction towards north illustrated in figure 1.
