Orbit modeling

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'or variations in gravitational potential around the surface of the Earth, the gravitational field of the Earth is modeled with spherical harmonics which are expressed through the equation:

${\mathbf {f} }=-{\frac {\mu }{R^{2}}}\mathbf {\hat {r}} +\sum _{n=2}^{\infty }\sum _{m=0}^{n}{\mathbf {f} }_{n,m}$

where

${\mu }$ is the gravitational parameter defined as the product of G, the universal gravitational constant, and the mass of the primary body.

$\mathbf {\hat {r}}$ is the unit vector defining the distance between the primary and secondary bodies, with ${R}$ being the magnitude of the distance.

${\mathbf {f} }_{n,m}$ represents the contribution to ${\mathbf {f} }$ of the spherical harmonic of degree n and order m, which is defined as:

${\begin{aligned}\mathbf {f} _{n,m}&={\frac {\mu R_{O}^{2}}{R^{n+m+1}}}\left({\frac {C_{n,m}{\mathcal {C}}_{m}+S_{n,m}{\mathcal {S}}_{m}}{R}}(A_{n,m+1}\mathbf {\hat {e}} _{3}-\left(s_{\lambda }A_{n,m+1}+(n+m+1)A_{n,m}\right)\mathbf {\hat {r}} \right)\\[10pt]&{}\quad {}+mA_{n,m}((C_{n,m}{\mathcal {C}}_{m-1}+S_{n,m}{\mathcal {S}}_{m-1})\mathbf {\hat {e}} _{1}+(S_{n,m}{\mathcal {C}}_{m-1}-C_{n,m}{\mathcal {S}}_{m-1})\mathbf {\hat {e}} _{2}))\end{aligned}}$

where:

$R_{O}$ is the mean equatorial radius of the primary body.

$R$ is the magnitude of the position vector from the center of the primary body to the center of the secondary body.

$C_{n,m}$ and $S_{n,m}$ are gravitational coefficients of degree n and order m. These are typically found through gravimetry measurements.

The unit vectors $\mathbf {\hat {e}} _{1},\mathbf {\hat {e}} _{2},\mathbf {\hat {e}} _{3}$ define a coordinate system fixed on the primary body. For the Earth, $\mathbf {\hat {e}} _{1}$ lies in the equatorial plane parallel to a line intersecting Earth's geometric center and the Greenwich meridian,$\mathbf {\hat {e}} _{3}$ points in the direction of the North polar axis, and $\mathbf {\hat {e}} _{2}=\mathbf {\hat {e}} _{3}\times \mathbf {\hat {e}} _{1}$

$A_{n,m}$ is referred to as a derived Legendre polynomial of degree n and order m. They are solved through the recurrence relation: $A_{n,m}(u)={\frac {1}{n-m}}((2n-1)uA_{n-1,m}(u)-(n+m-1)A_{n-2,m}(u))$

$s_{\lambda }$ is sine of the geographic latitude of the secondary body, which is $\mathbf {\hat {r}} \cdot \mathbf {\hat {e}} _{3}$.

${\mathcal {C}}_{m},{\mathcal {S}}_{m}$ are defined with the following recurrence relation and initial conditions:${\mathcal {C}}_{m}={\mathcal {C}}_{1}{\mathcal {C}}_{m-1}-{\mathcal {S}}_{1}{\mathcal {S}}_{m-1},{\mathcal {S}}_{m}={\mathcal {S}}_{1}{\mathcal {C}}_{m-1}+{\mathcal {C}}_{1}{\mathcal {S}}_{m-1},{\mathcal {S}}_{0}=0,{\mathcal {S}}_{1}=\mathbf {R} \cdot \mathbf {\hat {e}} _{2},{\mathcal {C}}_{0}=1,{\mathcal {C}}_{1}=\mathbf {R} \cdot \mathbf {\hat {e}} _{1}$

When modeling perturbations of an orbit around a primary body only the sum of the ${\mathbf {f} }_{n,m}$ terms need to be included in the perturbation since the point-mass gravity model is accounted for in the $-{\frac {\mu }{R^{2}}}\mathbf {\hat {r}}$ term

Third-body perturbations

Gravitational forces from third bodies can cause perturbations to an orbit. For example, the Sun and Moon cause perturbations to Orbits around the Earth. 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. 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:

${\dot {\Omega }}_{\mathrm {MOON} }=-0.00338(\cos(i))/n$

${\dot {\omega }}_{\mathrm {MOON} }=-0.00169(4-5\sin ^{2}(i))/n$

where:

${\dot {\Omega }}$ is the change to the right ascension of the ascending node in degrees per day.

${\dot {\omega }}$ is the change to the argument of perigee in degrees per day.

$i$ is the orbital inclination.

$n$ is the number of orbital revolutions per day.

Solar radiation

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:

$a_{R}\approx -4.5\times 10^{-6}(1+r)A/m$

where:

$a_{R}$ is the magnitude of acceleration in meters per second-squared.

$A$ is the cross-sectional area exposed to the Sun in meters-squared.

$m$ is the spacecraft mass in kilograms.

$r$ is the reflection factor which depends on material properties. $r=0$ for absorption, $r=1$ for specular reflection, and $r\approx 0.4$ for diffuse reflection.

For orbits around the Earth, solar radiation pressure becomes a stronger force than drag above 800 km altitude.

Propulsion

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:

$F_{n}={\dot {m}}\;v_{\text{e}}={\dot {m}}\;v_{\text{e-act}}+A_{\text{e}}(p_{\text{e}}-p_{\text{amb}})$

where:
${\dot {m}}$	=  exhaust gas mass flow
$v_{\text{e}}$	=  effective exhaust velocity
$v_{\text{e-act}}$	=  actual jet velocity at nozzle exit plane
$A_{\text{e}}$	=  flow area at nozzle exit plane (or the plane where the jet leaves the nozzle if separated flow)
$p_{\text{e}}$	=  static pressure at nozzle exit plane
$p_{\text{amb}}$	=  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. The perturbation model due to the solar wind can be used as a model of propulsive force from a solar sail.

Drag

Main article: Drag (physics)

The primary non-gravitational force acting on satellites in low Earth orbit is atmospheric drag. 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:

$F_{D}\,=\,{\tfrac {1}{2}}\,\rho \,v^{2}\,C_{d}\,A,$

where

$\mathbf {F} _{D}$ is the force of drag,

$\mathbf {} \rho$ is the density of the fluid,

$\mathbf {} v$ is the velocity of the object relative to the fluid,

$\mathbf {} C_{d}$ is the drag coefficient (a dimensionless parameter, e.g. 2 to 4 for most satellites)

$\mathbf {} A$ is the reference area.

Orbits with an altitude below 120 km 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 600 km have relatively small drag so that the orbit decays slow enough that it has no real impact on the satellite over its useful life. 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.

Magnetic fields

Magnetic fields can play a significant role as a source of orbit perturbation as was seen in the Long Duration Exposure Facility. Like gravity, the magnetic field of the Earth can be expressed through spherical harmonics as shown below:

${\mathbf {B} }=\sum _{n=1}^{\infty }\sum _{m=0}^{n}{\mathbf {B} }_{n,m}$

where

${\mathbf {B} }$ is the magnetic field vector at a point above the Earth's surface.

${\mathbf {B} }_{n,m}$ represents the contribution to ${\mathbf {B} }$ of the spherical harmonic of degree n and order m, defined as:

${\begin{aligned}\mathbf {B} _{n,m}={}&{\frac {K_{n,m}a^{n+2}}{R^{n+m+1}}}\left[{\frac {g_{n,m}{\mathcal {C}}_{m}+h_{n,m}{\mathcal {S}}_{m}}{R}}((s_{\lambda }A_{n,m+1}+(n+m+1)A_{n,m})\mathbf {\hat {r}} )-A_{n,m+1}\mathbf {\hat {e}} _{3}\right]\\[10pt]&{}-mA_{n,m}((g_{n,m}{\mathcal {C}}_{m-1}+h_{n,m}{\mathcal {S}}_{m-1})\mathbf {\hat {e}} _{1}+(h_{n,m}{\mathcal {C}}_{m-1}-g_{n,m}{\mathcal {S}}_{m-1})\mathbf {\hat {e}} _{2}))\end{aligned}}$

where:

$a$ is the mean equatorial radius of the primary body.

$R$ is the magnitude of the position vector from the center of the primary body to the center of the secondary body.

$\mathbf {\hat {r}}$ is a unit vector in the direction of the secondary body with its origin at the center of the primary body.

$g_{n,m}$ and $h_{n,m}$ are Gauss coefficients of degree n and order m. These are typically found through magnetic field measurements.

The unit vectors $\mathbf {\hat {e}} _{1},\mathbf {\hat {e}} _{2},\mathbf {\hat {e}} _{3}$ define a coordinate system fixed on the primary body. For the Earth, $\mathbf {\hat {e}} _{1}$ lies in the equatorial plane parallel to a line intersecting Earth's geometric center and the Greenwich meridian,$\mathbf {\hat {e}} _{3}$ points in the direction of the North polar axis, and $\mathbf {\hat {e}} _{2}=\mathbf {\hat {e}} _{3}\times \mathbf {\hat {e}} _{1}$

$A_{n,m}$ is referred to as a derived Legendre polynomial of degree n and order m. They are solved through the recurrence relation: $A_{n,m}(u)={\frac {1}{n-m}}((2n-1)uA_{n-1,m}(u)-(n+m-1)A_{n-2,m}(u))$

$K_{n,m}$ is defined as: 1 if m = 0, ${\big [}{\frac {n-m}{n+m}}{\big ]}^{0.5}K_{n-1,m}$ for $n\geq (m+1)$ and $m=[1\ldots \infty ]$ , and $[(n+m)(n-m+1)]^{-0.5}K_{n,m-1}$ for $n\geq m$ and $m=[2\ldots \infty ]$

$s_{\lambda }$ is sine of the geographic latitude of the secondary body, which is $\mathbf {\hat {r}} \cdot \mathbf {\hat {e}} _{3}$.

${\mathcal {C}}_{m},{\mathcal {S}}_{m}$ are defined with the following recurrence relation and initial conditions:${\mathcal {C}}_{m}={\mathcal {C}}_{1}{\mathcal {C}}_{m-1}-{\mathcal {S}}_{1}{\mathcal {S}}_{m-1},{\mathcal {S}}_{m}={\mathcal {S}}_{1}{\mathcal {C}}_{m-1}+{\mathcal {C}}_{1}{\mathcal {S}}_{m-1},{\mathcal {S}}_{0}=0,{\mathcal {S}}_{1}=\mathbf {R} \cdot \mathbf {\hat {e}} _{2},{\mathcal {C}}_{0}=1,{\mathcal {C}}_{1}=\mathbf {R} \cdot \mathbf {\hat {e}} _{1}$

See also

• Perturbation (astronomy)
• Osculating orbit
• Orbital resonance
• n-body problem
• two-body problem
• sphere of influence (astrodynamics)

External links

• [1] Gravity maps of the Earth