In orbital mechanics, a frozen orbit is an orbit for an artificial satellite in which perturbations have been minimized by careful selection of the orbital parameters. 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.[3] 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 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 (called "precession") 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 (97.8-99.0 degrees)[4] 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 (or vice versa). 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 (or vice versa). 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 (except for the orbital plane precession) 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 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. ... [There are] 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°.[6]
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.[8]
This analysis can be carried out as follows:
In the article orbital perturbation analysis the secular perturbation of the orbital pole
\Delta\hat{z}
J2
which can be expressed in terms of orbital elements thus:
Making a similar analysis for the
J3
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
J2
where:
Making the analysis for the
J3
For inclinations in the range 97.8–99.0 deg, the
\Delta\Omega
Adding the
J3
2\pi | J3 | |
\mu p3 |
3 | |
2 |
\sini \left(
5 | |
4 |
\sin2i - 1\right) (1, 0)
to one gets
Now the difference equation shows that the eccentricity vector will describe a circle centered at the point
\left( 0, - | J3 \sini |
J2 2 p |
\right)
-2\pi | J2 |
\mu p2 |
3\left(
5 | |
4 |
\sin2i - 1\right)
As
\mu=398600.440km3/s2
J2=1.7555 1010km5/s2
J3=-2.619 1011km6/s2
one gets for a polar orbit (
i=90\circ
p=7200km
(0, 0.001036)
The latter figure means that the eccentricity vector will have described a full circle in 1569 orbits.Selecting the initial mean eccentricity vector as
(0, 0.001036)
J2
J3
In terms of classical orbital elements, this means that a frozen orbit should have the following mean elements:
e=-
J3 \sini | |
J2 2 p |
\omega= 90\circ
The modern theory of frozen orbits is based on the algorithm given in a 1989 article by Mats Rosengren.[9]
For this the analytical expression 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. In this way the secular perturbation of the eccentricity vector caused by the
J2
J3
Applying this algorithm for the case discussed above, i.e. a polar orbit (
i=90\circ
p=7200km
J2
J3
(0, 0.001036)
When we also include the forces due to the higher zonal terms the optimal value changes to
(0, 0.001285)
Assuming in addition a reasonable solar pressure (a "cross-sectional-area" of, the direction to the sun in the direction towards the ascending node) the optimal value for the average eccentricity vector becomes
(0.000069, 0.001285)
\omega= 87\circ
\omega= 90\circ
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 "
J3
J3
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
Fr \hat{r}
Fλ \hat{λ}
\hat{λ}
\hat{r}
\hat{λ}
In the article Geopotential model it is shown that these force components caused by the
J3
To be able to apply relations derived in the article Orbital perturbation analysis (spacecraft) the force component
Fλ \hat{λ}
Ft \hat{t}
Fz \hat{z}
Let
\hat{a}, \hat{b}, \hat{n}
\hat{n}
\hat{a}, \hat{b}
\hat{a}
The components of the unit vectors
\hat{r}, \hat{t}, \hat{z}
making up the local coordinate system (of which
\hat{t}, \hat{z},
\hat{a}, \hat{b}, \hat{n}
ra=\cosu
rb=\cosi \sinu
rn=\sini \sinu
ta=-\sinu
tb=\cosi \cosu
tn=\sini \cosu
za=0
zb=-\sini
zn=\cosi
where
u
\hat{r}
\hat{g}=\hat{a}
\hat{h}=\cosi \hat{b} + \sini \hat{n}
Firstly
\sinλ= rn = \sini \sinu
where
λ
\hat{r}
Secondly the projection of direction north,
\hat{n}
\hat{t}, \hat{z},
\sini \cosu \hat{t} + \cosi \hat{z}
and this projection is
\cosλ \hat{λ}
where
\hat{λ}
\hat{λ}
From equation we see that
Fλ \hat{λ} =
-J | |||||
|
3 | |
2 |
2λ -1\right) \cosλ \hat{λ} = -J | |||||
\left(5 \sin | |||||
|
3 | |
2 |
\left(5 \sin2λ -1\right) (\sini \cosu \hat{t} + \cosi \hat{z})
and therefore:
In the article Orbital perturbation analysis (spacecraft) it is further shown that the secular perturbation of the orbital pole
\hat{z}
Introducing the expression for
Fz
The fraction
p | |
r |
p | |
r |
= 1+e ⋅ \cos\theta = 1+eg ⋅ \cosu+eh ⋅ \sinu
where
eg= e \cos\omega
eh= e \sin\omega
are the components of the eccentricity vector in the
\hat{g}, \hat{h}
As all integrals of type
2\pi | |
\int\limits | |
0 |
\cosmu \sinnu du
n
m
and
It follows that
where
\hat{g}
\hat{h}
\hat{g}
u
fz
\hat{z}
In the article Orbital perturbation analysis (spacecraft) it is shown that the secular perturbation of the eccentricity vector is
where
\hat{r},\hat{t}
\hat{r}
Vr=\sqrt{
\mu | |
p |
\hat{r}
Vt=\sqrt{
\mu | |
p |
\hat{t}
Introducing the expression for
Fr, Ft
Using that
Vr | |
Vt |
=
eg ⋅ \sinu - eh ⋅ \cosu | |||
|
the integral above can be split in 8 terms:
Given that
\hat{r}=\cosu \hat{g} + \sinu \hat{h}
\hat{t}=-\sinu \hat{g} + \cosu \hat{h}
we obtain
p | |
r |
= 1+e ⋅ \cos\theta = 1+eg ⋅ \cosu+eh ⋅ \sinu
and that all integrals of type
2\pi | |
\int\limits | |
0 |
\cosmu \sinnu du
n
m
Term 1
Term 2
Term 3
Term 4
Term 5
Term 6
Term 7
Term 8
As
It follows that