Orbital perturbation analysis is the activity of determining why a satellite's orbit differs from the mathematical ideal orbit. A satellite's orbit in an ideal two-body system describes a conic section, usually an ellipse. In reality, there are several factors that cause the conic section to continually change. These deviations from the ideal Kepler's orbit are called perturbations.
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Perturbation of spacecraft orbits
It has long been recognized that the Moon does not follow a perfect orbit, and many theories and models have been examined over the millennia to explain it. Isaac Newton determined the primary contributing factor to orbital perturbation of the moon was that the shape of the Earth is actually an oblate spheroid due to its spin, and he used the perturbations of the lunar orbit to estimate the oblateness of the Earth.
In Newton's Philosophiæ Naturalis Principia Mathematica, he demonstrated that the gravitational force between two mass points is inversely proportional to the square of the distance between the points, and he fully solved the corresponding "two-body problem" demonstrating that the radius vector between the two points would describe an ellipse. But no exact closed analytical form could be found for the three body problem. Instead, mathematical models called "orbital perturbation analysis" have been developed. With these techniques a quite accurate mathematical description of the trajectories of all the planets could be obtained. Newton recognized that the Moon's perturbations could not entirely be accounted for using just the solution to the three body problem, as the deviations from a pure Kepler orbit around the Earth are much larger than deviations of the orbits of the planets from their own Sun-centered Kepler orbits, caused by the gravitational attraction between the planets. With the availability of digital computers and the ease with which we can now compute orbits, this problem has partly disappeared, as the motion of all celestial bodies including planets, satellites, asteroids and comets can be modeled and predicted with almost perfect accuracy using the method of the numerical propagation of the trajectories. Nevertheless several analytical closed form expressions for the effect of such additional "perturbing forces" are still very useful.
All celestial bodies of the Solar System follow in first approximation a Kepler orbit around a central body. For a satellite (artificial or natural) this central body is a planet. But both due to gravitational forces caused by the Sun and other celestial bodies and due to the flattening of its planet (caused by its rotation which makes the planet slightly oblate and therefore the result of the Shell theorem not fully applicable) the satellite will follow an orbit around the Earth that deviates more than the Kepler orbits observed for the planets.
The precise modeling of the motion of the Moon has been a difficult task. The best and most accurate modeling for the lunar orbit before the availability of digital computers was obtained with the complicated Delaunay and Brown's lunar theories.
For man-made spacecraft orbiting the Earth at comparatively low altitudes the deviations from a Kepler orbit are much larger than for the Moon. The approximation of the gravitational force of the Earth to be that of a homogeneous sphere gets worse the closer one gets to the Earth surface and the majority of the artificial Earth satellites are in orbits that are only a few hundred kilometers over the Earth surface. Furthermore they are (as opposed to the Moon) significantly affected by the solar radiation pressure because of their large cross-section to mass ratio; this applies in particular to 3-axis stabilized spacecraft with large solar arrays. In addition they are significantly affected by rarefied air below 800–1000 km. The air drag at high altitudes is also dependent on solar activity.
Mathematical approach
Consider any function
of the position
and the velocity
From the chain rule of differentiation one gets that the time derivative of
where
If now
one has that
If the force is the sum of the "Kepler force" and an additional force (force per unit mass)
i.e.
one therefore has
and that the change of
If now the additional force
In general one wants to find an approximate expression for the change
This integral is evaluated setting
For the special case where the Kepler orbit is circular or almost circular
where
Perturbation of the semi-major axis/orbital period
For an elliptic Kepler orbit, the sum of the kinetic and the potential energy
where
If
and for a circular or almost circular orbit
From the change
Perturbation of the orbital plane
Let
where
For a circular or almost circular orbit (5) simplifies to
Example
In a circular orbit a low-force propulsion system (Ion thruster) generates a thrust (force per unit mass) of
The average change rate
where
Perturbation of the eccentricity vector
Rather than applying (1) and (2) on the partial derivatives of the orbital elements eccentricity and argument of perigee directly one should apply these relations for the eccentricity vector. First of all the typical application is a near-circular orbit. But there are also mathematical advantages working with the partial derivatives of the components of this vector also for orbits with a significant eccentricity.
Equations (60), (55) and (52) of the Kepler orbit article say that the eccentricity vector is
where
from which follows that
where
(Equations (18) and (19) of the Kepler orbit article)
The eccentricity vector is by definition always in the osculating orbital plane spanned by
with
corresponding to the rotation of the orbital plane
But in practice the in-plane change of the eccentricity vector is computed as
ignoring the out-of-plane force and the new eccentricity vector
is subsequently projected to the new orbital plane orthogonal to the new orbit normal
computed as described above.
Example
The Sun is in the orbital plane of a spacecraft in a circular orbit with radius
where
This means the eccentricity vector will gradually increase in the direction
The effect of the Earth flattening
In the article Geopotential model the modeling of the gravitational field as a sum of spherical harmonics is discussed. By far, the dominating term is the "J2-term". This is a "zonal term" and corresponding force is therefore completely in a longitudinal plane with one component
To be able to apply relations derived in the previous section the force component
Let
The components of the unit vectors
making up the local coordinate system (of which
where
Firstly
where
Secondly the projection of direction north,
and this projection is
where
From equation (12) of the article Geopotential model one therefore gets that
and therefore:
Perturbation of the orbital plane
From (5) and (20) one gets that
The fraction
where
As all integrals of type
are zero if not both
As
this can be written
As
In terms of orbital elements this is expressed as
where
Perturbation of the eccentricity vector
From (16), (18) and (19) follows that in-plane perturbation of the eccentricity vector is
the new eccentricity vector being the projection of
on the new orbital plane orthogonal to
where
Relative the coordinate system
one has that
Using that
and that
where
are the components of the eccentricity vector in the
This the difference equation of motion for the eccentricity vector
Translating this to orbital elements it must be remembered that the new eccentricity vector obtained by adding
This is illustrated in figure 3:
To the change in argument of the eccentricity vector
must be added an increment due to the precession of the orbital plane (caused by the out-of-plane force component) amounting to
One therefore gets that
In terms of the components of the eccentricity vector
where the first term is the in-plane perturbation of the eccentricity vector and the second is the effect of the new position of the ascending node in the new plane
From (28) follows that
Proof
Proof that the integral
where:
has the value
Integrating the first term of the integrand one gets:
and
For the second term one gets:
and
For the third term one gets:
and
For the fourth term one gets:
and
Adding the right hand sides of (32), (34), (36) and (38) one gets
Adding the right hand sides of (33), (35), (37) and (39) one gets