Orbit modeling is the process of creating mathematical models to simulate motion of a massive body as it moves in orbit around another massive body due to gravity. Other forces such as gravitational attraction from tertiary bodies, air resistance, solar pressure, or thrust from a propulsion system are typically modeled as secondary effects. Directly modeling an orbit can push the limits of machine precision due to the need to model small perturbations to very large orbits. Because of this, perturbation methods are often used to model the orbit in order to achieve better accuracy.
The study of orbital motion and mathematical modeling of orbits began with the first attempts to predict planetary motions in the sky, although in ancient times the causes remained a mystery. Newton, at the time he formulated his laws of motion and of gravitation, applied them to the first analysis of perturbations, recognizing the complex difficulties of their calculation.[1] Many of the great mathematicians since then have given attention to the various problems involved; throughout the 18th and 19th centuries there was demand for accurate tables of the position of the Moon and planets for purposes of navigation at sea.
The complex motions of orbits can be broken down. The hypothetical motion that the body follows under the gravitational effect of one other body only is typically a conic section, and can be readily modeled with the methods of geometry. This is called a two-body problem, or an unperturbed Keplerian orbit. The differences between the Keplerian orbit and the actual motion of the body are caused by perturbations. These perturbations are caused by forces other than the gravitational effect between the primary and secondary body and must be modeled to create an accurate orbit simulation. Most orbit modeling approaches model the two-body problem and then add models of these perturbing forces and simulate these models over time. Perturbing forces may include gravitational attraction from other bodies besides the primary, solar wind, drag, magnetic fields, and propulsive forces.
Analytical solutions (mathematical expressions to predict the positions and motions at any future time) for simple two-body and three-body problems exist; none have been found for the n-body problem except for certain special cases. Even the two-body problem becomes insoluble if one of the bodies is irregular in shape.[2]
Due to the difficulty in finding analytic solutions to most problems of interest, computer modeling and simulation is typically used to analyze orbital motion. A wide variety of software is available to simulate orbits and trajectories of spacecraft.
See main article: Kepler orbit. In its simplest form, an orbit model can be created by assuming that only two bodies are involved, both behave as spherical point-masses, and that no other forces act on the bodies. For this case the model is simplified to a Kepler orbit.
Keplerian orbits follow conic sections. The mathematical model of the orbit which gives the distance between a central body and an orbiting body can be expressed as:
r(\nu)=
a(1-e2) | |
1+e\cos(\nu) |
Where:
r
a
e
\nu
r(\nu)=
p | |
1+e\cos(\nu) |
Where
p
An alternate approach uses Isaac Newton's law of universal gravitation as defined below:
F=G
m1m2 | |
r2 |
where:
F
G
m1
m2
r
Making an additional assumption that the mass of the primary body is much greater than the mass of the secondary body and substituting in Newton's second law of motion, results in the following differential equation
\ddot{r
Solving this differential equation results in Keplerian motion for an orbit.In practice, Keplerian orbits are typically only useful for first-order approximations, special cases, or as the base model for a perturbed orbit.
See main article: Perturbation (astronomy). Orbit models are typically propagated in time and space using special perturbation methods. This is performed by first modeling the orbit as a Keplerian orbit. Then perturbations are added to the model to account for the various perturbations that affect the orbit.[1] Special perturbations can be applied to any problem in celestial mechanics, as it is not limited to cases where the perturbing forces are small.[2] Special perturbation methods are the basis of the most accurate machine-generated planetary ephemerides.[1] see, for instance, Jet Propulsion Laboratory Development Ephemeris
Cowell's method is a special perturbation method;[3] mathematically, for
n
i
j
\ddot{r
where
\ddot{r
i
G
mj
j
ri
rj
i
j
rij
i
j
x
y
z
Encke's method begins with the osculating orbit as a reference and integrates numerically to solve for the variation from the reference as a function of time.[6] Its advantages are that perturbations are generally small in magnitude, so the integration can proceed in larger steps (with resulting lesser errors), and the method is much less affected by extreme perturbations than Cowell's method. Its disadvantage is complexity; it cannot be used indefinitely without occasionally updating the osculating orbit and continuing from there, a process known as rectification.[4] [7]
Letting
\boldsymbol{\rho}
r
\deltar
\ddot{r
\boldsymbol{\ddot{\rho}}
r
\boldsymbol{\rho}
where
\mu=G(M+m)
M
m
aper
r
\rho
r
\boldsymbol{\rho}
Substituting from equations and into equation,
which, in theory, could be integrated twice to find
\deltar
\boldsymbol{\rho}
\deltar
r
{\boldsymbol{\rho}\over\rho3}-{r\overr3}
In 1991 Victor R. Bond and Michael F. Fraietta created an efficient and highly accurate method for solving the two-body perturbed problem.[10] This method uses the linearized and regularized differential equations of motion derived by Hans Sperling and a perturbation theory based on these equations developed by C.A. Burdet in the year 1864. In 1973, Bond and Hanssen improved Burdet's set of differential equations by using the total energy of the perturbed system as a parameter instead of the two-body energy and by reducing the number of elements to 13. In 1989 Bond and Gottlieb embedded the Jacobian integral, which is a constant when the potential function is explicitly dependent upon time as well as position in the Newtonian equations. The Jacobian constant was used as an element to replace the total energy in a reformulation of the differential equations of motion. In this process, another element which is proportional to a component of the angular momentum is introduced. This brought the total number of elements back to 14. In 1991, Bond and Fraietta made further revisions by replacing the Laplace vector with another vector integral as well as another scalar integral which removed small secular terms which appeared in the differential equations for some of the elements.[11]
The Sperling–Burdet method is executed in a 5 step process as follows:[11]
Step 1: Initialization
Given an initial position,
r0
v0
t0
s=0
r0=(r0 ⋅ r
1/2 | |
0) |
a=r0
b=r0 ⋅ v0
\tau=t0
\boldsymbol{\alpha}=r0
\boldsymbol{\beta}=av0
Perturbations due to perturbing masses, defined as
V0
[{\partial{V}\over{\partial{r
Perturbations due to other accelerations, defined as
P0
\alpha | ||||
|
-v0 ⋅ v0-2V0
\gamma=\mu-\alphaJa
\boldsymbol{\delta}=-(v0 ⋅ v0)r0+(r0 ⋅ v0)v
|
r0-\alphaJr0
\sigma=0
Step 2: Transform elements to coordinates
2c | |
r=\boldsymbol{\alpha}+\boldsymbol{\beta}sc | |
2 |
r'=\boldsymbol{\beta}c0+\boldsymbol{\delta}sc1
x3=\alphaJ(\boldsymbol{\alpha}-r)+\boldsymbol{\delta}
\gamma=\mu-\alphaJa
r=a+bsc1+\gamma
2c | |
s | |
2 |
v=r'/r
r'=bc0+\gammasc1
2c | |
t=\tau+as+bs | |
2+\gamma |
3c | |
s | |
3 |
where
c0,c1,c2,c3
Step 3: Evaluate differential equations for the elements
F=P-{\partial{V}\over\partial{r
Q=r2F+2r(-V+\sigma)
\alpha'J=2(-r'+r\boldsymbol{\omega} x r) ⋅ P
\mu\boldsymbol{\epsilon}'=2(r' ⋅ F)r-(r ⋅ F)r'-(r ⋅ r')F
2c | |
\boldsymbol{\alpha}'=-Qsc | |
2-\alpha' |
3\bar{c} | ||||
|
\boldsymbol{\delta}s4c
2 | |
2] |
\boldsymbol{\beta}'=Qc0+\mu\boldsymbol{\epsilon}'sc1+\alpha'J[\boldsymbol{\alpha}sc
3(2\bar{c} | |
3-c |
1c2)]
\boldsymbol{\delta}'=Q\alphaJsc1-\mu\boldsymbol{\epsilon}'c0+\alpha'J[-\boldsymbol{\alpha}c0+2\alpha
3\bar{c} | ||||
|
4c | |
\boldsymbol{\delta}\alpha | |
Js |
2 | |
2] |
\sigma'=r\boldsymbol{\omega} ⋅ r x F
a'=- | 1 |
r |
r ⋅ Qsc1-\alpha
3\bar{c} | ||||
|
\gammas4c
2 | |
2] |
b'= | 1 |
r |
r ⋅ Qc0+\alphaJ'[asc
2\bar{c} | |
2-\gamma |
3(2\bar{c} | |
s | |
3-c |
1c2)]
\gamma'=- | 1 |
r |
r ⋅ Q\alphaJsc1+\alphaJ'[-ac0+2b\alpha
3\bar{c} | ||||
|
\gamma\alphaJs4c
2 | |
2] |
\tau'= | 1 |
r |
2c | |
r ⋅ Qs | |
2+\alpha |
3c | ||||
|
bs4c
2 | |
2-2\gamma |
5(c | |
s | |
5-4\bar{c} |
5)]
Step 4: Integration
Here the differential equations are integrated over a period
\Deltas
s+\Deltas
Step 5: Advance
Set
s=s+\Deltas
See main article: Perturbation (astronomy).
Perturbing forces cause orbits to become perturbed from a perfect Keplerian orbit. Models for each of these forces are created and executed during the orbit simulation so their effects on the orbit can be determined.
The Earth is not a perfect sphere nor is mass evenly distributed within the Earth. This results in the point-mass gravity model being inaccurate for orbits around the Earth, particularly Low Earth orbits. To account for variations in gravitational potential around the surface of the Earth, the gravitational field of the Earth is modeled with spherical harmonics[12] which are expressed through the equation:
{f
{\mu}
\hat{r
{R}
{f
{f
\begin{align} fn,m&=
| \left( | |||||||||
Rn+m+1 |
Cn,ml{C | |
m+S |
n,ml{S}m}{R}(An,m+1\hat{e
where:
RO
R
Cn,m
Sn,m
The unit vectors
\hat{e
\hat{e
\hat{e
\hat{e
An,m
An,m(u)=
1 | |
n-m |
((2n-1)uAn-1,m(u)-(n+m-1)An-2,m(u))
sλ
\hat{r
l{C}m,l{S}m
l{C}m=l{C}1l{C}m-1-l{S}1l{S}m-1,l{S}m=l{S}1l{C}m-1+l{C}1l{S}m-1,l{S}0=0,l{S}1=R ⋅ \hat{e
{f
- | \mu |
R2 |
\hat{r
Gravitational forces from third bodies can cause perturbations to an orbit. For example, the Sun and Moon cause perturbations to Orbits around the Earth.[13] These forces are modeled in the same way that gravity is modeled for the primary body by means of direct gravitational N-body simulations. Typically, only a spherical point-mass gravity model is used for modeling effects from these third bodies.[14] Some special cases of third-body perturbations have approximate analytic solutions. For example, perturbations for the right ascension of the ascending node and argument of perigee for a circular Earth orbit are:[13]
\Omega |
MOON=-0.00338(\cos(i))/n
\omega |
2(i))/n | |
MOON=-0.00169(4-5\sin |
where:
\Omega |
\omega |
i
n
See main article: Radiation pressure. Solar radiation pressure causes perturbations to orbits. The magnitude of acceleration it imparts to a spacecraft in Earth orbit is modeled using the equation below:[13]
aR ≈ -4.5 x 10-6(1+r)A/m
aR
A
m
r
r=0
r=1
r ≈ 0.4
For orbits around the Earth, solar radiation pressure becomes a stronger force than drag above 800km (500miles) altitude.[13]
See main article: Spacecraft propulsion. There are many different types of spacecraft propulsion. Rocket engines are one of the most widely used. The force of a rocket engine is modeled by the equation:[15]
Fn=
m |
ve=
m |
ve-act+Ae(pe-pamb)
where: | |||
| = exhaust gas mass flow | ||
---|---|---|---|
ve | = effective exhaust velocity | ||
ve-act | = actual jet velocity at nozzle exit plane | ||
Ae | = flow area at nozzle exit plane (or the plane where the jet leaves the nozzle if separated flow) | ||
pe | = static pressure at nozzle exit plane | ||
pamb | = ambient (or atmospheric) pressure |
Another possible method is a solar sail. Solar sails use radiation pressure in a way to achieve a desired propulsive force.[16] The perturbation model due to the solar wind can be used as a model of propulsive force from a solar sail.
See main article: Drag (physics). The primary non-gravitational force acting on satellites in low Earth orbit is atmospheric drag.[13] Drag will act in opposition to the direction of velocity and remove energy from an orbit. The force due to drag is modeled by the following equation:
FD=\tfrac12\rhov2CdA,
where
FD
\rho
v
Cd
A
Orbits with an altitude below 120km (80miles) generally have such high drag that the orbits decay too rapidly to give a satellite a sufficient lifetime to accomplish any practical mission. On the other hand, orbits with an altitude above 600km (400miles) have relatively small drag so that the orbit decays slow enough that it has no real impact on the satellite over its useful life.[13] Density of air can vary significantly in the thermosphere where most low Earth orbiting satellites reside. The variation is primarily due to solar activity, and thus solar activity can greatly influence the force of drag on a spacecraft and complicate long-term orbit simulation.[13]
Magnetic fields can play a significant role as a source of orbit perturbation as was seen in the Long Duration Exposure Facility.[12] Like gravity, the magnetic field of the Earth can be expressed through spherical harmonics as shown below:[12]
{B
{B
{B
{B
\begin{align} Bn,m={}&
Kn,man+2 | \left[ | |
Rn+m+1 |
gn,ml{C | |
m+h |
n,ml{S}m}{R}((sλAn,m+1+(n+m+1)An,m)\hat{r
where:
a
R
\hat{r
gn,m
hn,m
The unit vectors
\hat{e
\hat{e
\hat{e
\hat{e
An,m
An,m(u)=
1 | |
n-m |
((2n-1)uAn-1,m(u)-(n+m-1)An-2,m(u))
Kn,m
[ | n-m |
n+m |
]0.5Kn-1,m
n\ge(m+1)
m=[1\ldotsinfty]
[(n+m)(n-m+1)]-0.5Kn,m-1
n\gem
m=[2\ldotsinfty]
sλ
\hat{r
l{C}m,l{S}m
l{C}m=l{C}1l{C}m-1-l{S}1l{S}m-1,l{S}m=l{S}1l{C}m-1+l{C}1l{S}m-1,l{S}0=0,l{S}1=R ⋅ \hat{e