In
orbital mechanics
Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of ...
, a frozen orbit is an
orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
for an artificial
satellite
A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioisotope ...
in which natural drifting due to the
central body
A primary (also called a gravitational primary, primary body, or central body) is the main physical body of a gravitationally bound, multi-object system. This object constitutes most of that system's mass and will generally be located near the syst ...
's shape has been minimized by careful selection of the
orbital parameters
Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. There are many different ways to mathematically describe the same ...
. Typically, this is an orbit in which, over a long period of time, the satellite's
altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
remains constant at the same point in each orbit.
Changes in the
inclination
Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a Plane of reference, reference plane and the orbital plane or Axis of rotation, axis of direction of the orbiting object ...
,
position
Position often refers to:
* Position (geometry), the spatial location (rather than orientation) of an entity
* Position, a job or occupation
Position may also refer to:
Games and recreation
* Position (poker), location relative to the dealer
* ...
of the
apsis
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two apsides in any ellip ...
of the orbit, and
eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-Centre (geometry), center, in geometry
* Eccentricity (g ...
have been minimized by choosing
initial value
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or ot ...
s so that their
perturbations cancel out.
, which 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
Radiation pressure is the mechanical pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of any wavelength that is a ...
, and
air drag
In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding ...
. 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 In celestial mechanics, the eccentricity vector of a Kepler orbit is the dimensionless vector with direction pointing from apoapsis to periapsis and with magnitude equal to the orbit's scalar eccentricity. For Kepler orbits the eccentricity vec ...
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 satellite
An Earth observation satellite or Earth remote sensing satellite is a satellite used or designed for Earth observation (EO) from orbit, including spy satellites and similar ones intended for non-military uses such as environmental monitoring, me ...
s
ERS-1, ERS-2 and
Envisat are operated in sun-synchronous frozen orbits.
Lunar frozen orbits
Through a study of many
lunar orbit
In astronomy, lunar orbit (also known as a selenocentric orbit) is the orbit of an object around the Moon.
As used in the space program, this refers not to the orbit of the Moon about the Earth, but to orbits by spacecraft around the Moon. The ...
ing satellites, scientists have discovered that most
low lunar orbits (LLO) are unstable.
[Frozen Orbits About the Moon. 2003]
/ref> Four ''frozen lunar orbits'' have been identified at 27°, 50°, 76°, and 86° inclination. NASA expounded on this in 2006:
Lunar mascons
In astronomy, astrophysics and geophysics, a mass concentration (or mascon) is a region of a planet's or moon's crust that contains a large positive gravity anomaly. In general, the word "mascon" can be used as a noun to refer to an excess distr ...
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. ... here are
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here
Television
* Here TV (formerly "here!"), a TV ...
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
Apollo 15 (July 26August 7, 1971) was the ninth crewed mission in the United States' Apollo program and the fourth to land on the Moon. It was the first J mission, with a longer stay on the Moon and a greater focus on science than ear ...
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
Apollo 16 (April 1627, 1972) was the tenth crewed mission in the United States Apollo space program, administered by NASA, and the fifth and penultimate to land on the Moon. It was the second of Apollo's " J missions", with an extended sta ...
had an orbital inclination of only 11°.
Classical theory
The classical theory of frozen orbits is essentially based on the analytical perturbation analysis
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
for artificial satellites of Dirk Brouwer
Dirk Brouwer (; September 1, 1902 – January 31, 1966) was a Dutch-American astronomer.
He received his PhD in 1927 at Leiden University under Willem de Sitter and then went to Yale University. From 1941 until 1966 he was editor of the ''Ast ...
made under contract with NASA
The National Aeronautics and Space Administration (NASA ) is an independent agency of the US federal government responsible for the civil space program, aeronautics research, and space research.
NASA was established in 1958, succeeding t ...
and published in 1959.
This analysis can be carried out as follows:
In the article orbital perturbation analysis
In astronomy, perturbation is the complex motion of a massive body subjected to forces other than the gravitational attraction of a single other massive body. The other forces can include a third (fourth, fifth, etc.) body, resistance, as from ...
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 (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 In celestial mechanics, the eccentricity vector of a Kepler orbit is the dimensionless vector with direction pointing from apoapsis to periapsis and with magnitude equal to the orbit's scalar eccentricity. For Kepler orbits the eccentricity vec ...
caused by the is shown to be:
where:
* The first term is the in-plane perturbation of the eccentricity vector caused by the in-plane component of the perturbing force
* The second term is the effect of the new position of the ascending node in the new orbital plane, the orbital plane being perturbed by the out-of-plane force component
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 (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
Radiation pressure is the mechanical pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of any wavelength that is a ...
, 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 , 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 satellite
An Earth observation satellite or Earth remote sensing satellite is a satellite used or designed for Earth observation (EO) from orbit, including spy satellites and similar ones intended for non-military uses such as environmental monitoring, me ...
s 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 (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
An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations ...
) such that points in the direction north and such that are in the equatorial plane of the Earth with pointing towards the ascending node
An orbital node is either of the two points where an orbit intersects a plane of reference to which it is inclined. A non-inclined orbit, which is contained in the reference plane, has no nodes.
Planes of reference
Common planes of refere ...
, i.e. towards the blue point of Figure 2.
The components of the unit vectors
:
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
Firstly
:
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
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 (spacecraft) 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
: and are the base vectors of the rectangular coordinate system in the plane of the reference Kepler orbit with in the equatorial plane towards the ascending node and is the polar argument relative this equatorial coordinate system
: is the force component (per unit mass) in the direction of the orbit pole
In the article Orbital perturbation analysis (spacecraft) it is shown that the secular perturbation of the eccentricity vector is
where
* is the usual local coordinate system with unit vector directed away from the Earth
* - the velocity component in direction
* - the velocity component in direction
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
References
Further reading
STUDY OF ORBITAL ELEMENTS ON THE NEIGHBOURHOOD OF A FROZEN ORBIT.
including atmospheric drag and J5 terms
{{DEFAULTSORT:Frozen Orbit
Astrodynamics
Orbits
Spaceflight concepts