Frozen orbit


In orbital mechanics, a frozen orbit is an orbit for an artificial satellite in which natural drifting due to the central body's shape has been minimized by careful selection of the orbital parameters. Typically, this is an orbit in which, over a long period of time, the satellite's altitude remains constant at the same point in each orbit. Changes in the inclination, position of the lowest point of the orbit, and eccentricity have been minimized by choosing initial values so that their perturbations cancel out. This results in a long-term stable orbit that minimizes the use of station-keeping propellant.

Background and motivation

For most spacecraft, changes to orbits are caused by the oblateness of the Earth, gravitational attraction from the sun and moon, solar radiation pressure, and air drag. These are called "perturbing forces". They must be counteracted by maneuvers to keep the spacecraft in the desired orbit. For a geostationary spacecraft, correction maneuvers on the order of 40–50 m/s 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.
For sun-synchronous spacecraft, intentional shifting of the orbit plane can be used for the benefit of the mission. For these missions, a near-circular orbit with an altitude of 600–900 km is used. An appropriate inclination is selected so that the precession of the orbital plane is equal to the rate of movement of the Earth around the sun, about 1 degree per day.
As a result, the spacecraft will pass over points on the Earth that have the same time of day during every orbit. For instance, if the orbit is "square to the sun", the vehicle will always pass over points at which it is 6 a.m. on the north-bound portion, and 6 p.m. on the south-bound portion. This is called a "Dawn-Dusk" 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. These are called "Noon-Midnight" orbits. Such orbits are desirable for many Earth observation missions such as weather, imagery, and mapping.
The perturbing force caused by the oblateness of the Earth will in general perturb not only the orbital plane but also the eccentricity vector of the orbit. There exists, however, an almost circular orbit for which there are no secular/long periodic perturbations of the eccentricity vector, only periodic perturbations with period equal to the orbital period. Such an orbit is then perfectly periodic and it is therefore called a "frozen orbit". Such an orbit is often the preferred choice for an Earth observation mission where repeated observations of the same area of the Earth should be made under as constant observation conditions as possible.
The Earth observation satellites ERS-1, ERS-2 and Envisat are operated in sun-synchronous frozen orbits.

Lunar frozen orbits

Through a study of many lunar orbiting satellites, scientists have discovered that most low lunar orbits are unstable. Four frozen lunar orbits have been identified at 27°, 50°, 76°, and 86° inclination. NASA expounded on 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 will eventually crash into the Moon.... a number of 'frozen orbits' where a spacecraft can stay in a low lunar orbit indefinitely. They occur at four inclinations: 27°, 50°, 76°, and 86°"—the last one being nearly over the lunar poles. The orbit of the relatively long-lived Apollo 15 subsatellite PFS-1 had an inclination of 28°, which turned out to be close to the inclination of one of the frozen orbits—but less fortunate PFS-2 had an orbital inclination of only 11°.

Classical theory

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.
This analysis can be carried out as follows:
In the article orbital perturbation analysis the secular perturbation of the orbital pole from the term of the geopotential model is shown to be
which can be expressed in terms of orbital elements thus:
Making a similar analysis for the term, 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 is shown to be:
where:
Making the analysis for the term one gets for the first term, i.e. for the perturbation of the eccentricity vector from the in-plane force component
For inclinations in the range 97.8–99.0 deg, the value given by is much smaller than the value given by and can be ignored. Similarly the quadratic terms of the eccentricity vector components in can be ignored for almost circular orbits, i.e. can be approximated with
Adding the contribution
to one gets
Now the difference equation shows that the eccentricity vector will describe a circle centered at the point ; the polar argument of the eccentricity vector increases with radians between consecutive orbits.
As
one gets for a polar orbit with that the centre of the circle is at and the change of polar argument is 0.00400 radians per orbit.
The latter figure means that the eccentricity vector will have described a full circle in 1569 orbits.
Selecting the initial mean eccentricity vector as the mean eccentricity vector will stay constant for successive orbits, i.e. the orbit is frozen because the secular perturbations of the term given by and of the term given by cancel out.
In terms of classical orbital elements, this means that a frozen orbit should have the following mean elements:

Modern theory

The modern theory of frozen orbits is based on the algorithm given in a 1989 article by Mats Rosengren.
For this the analytical expression is used to iteratively update the initial eccentricity vector to obtain that the eccentricity vector several orbits later computed by the precise numerical propagation takes precisely the same value. In this way the secular perturbation of the eccentricity vector caused by the term is used to counteract all secular perturbations, not only those caused by the term. One such additional secular perturbation that in this way can be compensated for is the one caused by the solar radiation pressure, this perturbation is discussed in the article "Orbital perturbation analysis ".
Applying this algorithm for the case discussed above, i.e. a polar orbit with ignoring all perturbing forces other than the and the 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 the optimal value for the average eccentricity vector becomes which corresponds to :, i.e. the optimal value is not anymore.
This algorithm is implemented in the orbit control software used for the Earth observation satellites ERS-1, ERS-2 and Envisat

Derivation of the closed form expressions for the ''J''3 perturbation

The main perturbing force to be counteracted in order to have a frozen orbit is the " force", i.e. the gravitational force caused by an imperfect symmetry north/south of the Earth, and the "classical theory" is based on the closed form expression for this " perturbation". With the "modern theory" this explicit closed form expression is not directly used but it is certainly still worthwhile to derive it.
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 radial direction and one component with the unit vector orthogonal to the radial direction towards north. These directions and are illustrated in Figure 1.
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 the force component must be split into two orthogonal components and as illustrated in figure 2
Let make up a rectangular coordinate system with origin in the center of the Earth such that points in the direction north and such that are in the equatorial plane of the Earth with pointing towards the ascending node, i.e. towards the blue point of Figure 2.
The components of the unit vectors
making up the local coordinate system, and expressing their relation with, are as follows:
where is the polar argument of relative the orthogonal unit vectors and in the orbital plane
Firstly
where is the angle between the equator plane and and from equation of the article Geopotential model one therefore obtains
Secondly the projection of direction north,, on the plane spanned by is
and this projection is
where is the unit vector orthogonal to the radial direction towards north illustrated in figure 1.
From equation we see that
and therefore:
In the article Orbital perturbation analysis it is further shown that the secular perturbation of the orbital pole is
Introducing the expression for of in one gets
The fraction is
where
are the components of the eccentricity vector in the coordinate system.
As all integrals of type
are zero if not both and are even, we see that
and
It follows that
where
In the article Orbital perturbation analysis it is shown that the secular perturbation of the eccentricity vector is
where
Introducing the expression for of and in one gets
Using that
the integral above can be split in 8 terms:
Given that
we obtain
and that all integrals of type
are zero if not both and are even:
Term 1
Term 2
Term 3
Term 4
Term 5
Term 6
Term 7
Term 8
As
It follows that