“Frozen orbit”的意思、由来-开放百科全书

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.^[1] 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.^[2] 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 (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) 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 frozen orbits

Through a study of many lunar orbiting satellites, scientists have discovered that most low lunar orbits (LLO) are unstable.^[3] Four frozen lunar orbits have been identified at 27°, 50°, 76°, and 86° inclination. NASA expounded on this in 2006:

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.^[5]

In the article orbital perturbation analysis the secular perturbation of the orbital pole

from the

term of the geopotential model is shown to be

Making a similar analysis for the

term (corresponding to the fact that the earth is slightly pear shaped), one gets

In the same article the secular perturbation of the components of the eccentricity vector caused by the

is shown to be:

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 ({{EquationNote|6}}) is much smaller than the value given by ({{EquationNote|3}}) and can be ignored. Similarly the quadratic terms of the eccentricity vector components in ({{EquationNote|8}}) can be ignored for almost circular orbits, i.e. ({{EquationNote|8}}) can be approximated with

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.

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 ({{EquationNote|7}}) and of the

term given by ({{EquationNote|9}}) 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.^[6]

For this the analytical expression ({{EquationNote|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. In this way the secular perturbation of the eccentricity vector caused by the

term is used to counteract all secular perturbations, not only those (dominating) 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 (spacecraft)".

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 (a "cross-sectional-area" of {{nowrap|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

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 J₃ perturbation

The main perturbing force to be counter-acted 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 potential from a zonal term is rotational symmetric around the polar axis of the Earth and corresponding force is entirely in a longitudial 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 (spacecraft) 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 (in the center of the Reference ellipsoid) 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.

making up the local coordinate system (of which

are illustrated in figure 2), and expressing their relation with

, are as follows:

where

is the polar argument of

relative the orthogonal unit vectors

and

in the orbital plane

where

is the angle between the equator plane and

(between the green points of figure 2) and from equation (12) of the article Geopotential model one therefore obtains

where

is the unit vector

orthogonal to the radial direction towards north illustrated in figure 1.

In the article Orbital perturbation analysis (spacecraft) it is further shown that the secular perturbation of the orbital pole

Introducing the expression for

of ({{EquationNote|14}}) in ({{EquationNote|15}}) one gets

In the article Orbital perturbation analysis (spacecraft) it is shown that the secular perturbation of the eccentricity vector is

Introducing the expression for

of ({{EquationNote|12}}) and ({{EquationNote|13}}) in ({{EquationNote|20}}) one gets

References

1. ^{{cite web|last=Eagle |first=C. David |title=Frozen Orbit Design |url=http://www.cdeagle.com/omnum/pdf/frozen1.pdf |work=Orbital Mechanics with Numerit |accessdate=5 April 2012 |deadurl=yes |archiveurl=https://web.archive.org/web/20111121072936/http://www.cdeagle.com/omnum/pdf/frozen1.pdf |archivedate=21 November 2011 |df=}}
2. ^{{cite book |last=Chobotov |first=Vladimir A. |title=Orbital Mechanics |edition=3rd |year=2002 |publisher=American Institute of Aeronautics and Astronautics |page=221 |url=http://www.knovel.com/web/portal/browse/display?_EXT_KNOVEL_DISPLAY_bookid=1586&VerticalID=0}}
3. ^[https://arc.aiaa.org/doi/abs/10.2514/2.5064?journalCode=jgcd Frozen Orbits About the Moon. 2003]
4. ^{{cite web |title=Bizarre Lunar Orbits |url=https://science.nasa.gov/science-news/science-at-nasa/2006/06nov_loworbit/ |last=Bell |first=Trudy E. |date=November 6, 2006 |editor-last=Phillips |editor-first=Tony |work=Science@NASA |publisher=NASA |accessdate=2017-09-08}}
5. ^Dirk Brouwer: "Solution of the Problem of the Artificial Satellite Without Drag", Astronomical Journal, 64 (1959)
6. ^{{cite conference |bibcode=1989ommd.proc...49R |author=Mats Rosengren |title=Improved technique for Passive Eccentricity Control (AAS 89-155) |volume=69 |publisher=AAS/NASA |book-title=Advances in the Astronautical Sciences|year=1989}}

Background and motivation

Lunar frozen orbits

Classical theory

Modern theory

Derivation of the closed form expressions for the J3 perturbation

References

Further reading

Derivation of the closed form expressions for the J₃ perturbation