Perturbation (astronomy)

In methods of general perturbations, general differential equations, either of motion or of change in the orbital elements, are solved analytically, usually by series expansions. The result is usually expressed in terms of algebraic and trigonometric functions of the orbital elements of the body in question and the perturbing bodies. This can be applied generally to many different sets of conditions, and is not specific to any particular set of gravitating objects.[5] Historically, general perturbations were investigated first. The classical methods are known as variation of the elements, variation of parameters or variation of the constants of integration. In these methods, it is considered that the body is always moving in a conic section, however the conic section is constantly changing due to the perturbations. If all perturbations were to cease at any particular instant, the body would continue in this (now unchanging) conic section indefinitely; this conic is known as the osculating orbit and its orbital elements at any particular time are what are sought by the methods of general perturbations.[2]

General perturbations takes advantage of the fact that in many problems of celestial mechanics, the two-body orbit changes rather slowly due to the perturbations; the two-body orbit is a good first approximation. General perturbations is applicable only if the perturbing forces are about one order of magnitude smaller, or less, than the gravitational force of the primary body.[4] In the Solar System, this is usually the case; Jupiter, the second largest body, has a mass of about 1/1000 that of the Sun.

General perturbation methods are preferred for some types of problems, as the source of certain observed motions are readily found. This is not necessarily so for special perturbations; the motions would be predicted with similar accuracy, but no information on the configurations of the perturbing bodies (for instance, an orbital resonance) which caused them would be available.[4]

In methods of special perturbations, numerical datasets, representing values for the positions, velocities and accelerative forces on the bodies of interest, are made the basis of numerical integration of the differential equations of motion.[6] In effect, the positions and velocities are perturbed directly, and no attempt is made to calculate the curves of the orbits or the orbital elements.[2]

Special perturbations can be applied to any problem in celestial mechanics, as it is not limited to cases where the perturbing forces are small.[4] Once applied only to comets and minor planets, special perturbation methods are now the basis of the most accurate machine-generated planetary ephemerides of the great astronomical almanacs.[2][7] Special perturbations are also used for modeling an orbit with computers.

Cowell's method. Forces from all perturbing bodies (black and gray) are summed to form the total force on body ${\displaystyle \ i\ }$ (red), and this is numerically integrated starting from the initial position (the epoch of osculation).

Cowell's formulation (so named for Philip H. Cowell, who, with A.C.D. Cromellin, used a similar method to predict the return of Halley's comet) is perhaps the simplest of the special perturbation methods.[8] In a system of ${\displaystyle \ n\ }$ mutually interacting bodies, this method mathematically solves for the Newtonian forces on body ${\displaystyle \ i\ }$ by summing the individual interactions from the other ${\displaystyle j}$ bodies:

${\displaystyle \mathbf {\ddot {r}} _{i}=\sum _{\underset {j\neq i}{j=1}}^{n}\ G\ m_{j}{\frac {\ (\mathbf {r} _{j}-\mathbf {r} _{i})\ }{\ \|\mathbf {r} _{j}-\mathbf {r} _{i}\|^{3}}}}$

where ${\displaystyle \ \mathbf {\ddot {r}} _{i}\ }$ is the acceleration vector of body ${\displaystyle i}$, ${\displaystyle G}$ is the gravitational constant, ${\displaystyle \ m_{j}\ }$ is the mass of body ${\displaystyle j}$, ${\displaystyle \ \mathbf {r} _{i}\ }$ and ${\displaystyle \ \mathbf {r} _{j}\ }$ are the position vectors of objects ${\displaystyle \ i\ }$ and ${\displaystyle \ j\ }$ respectively, and ${\displaystyle \ r_{ij}\equiv \|\mathbf {r} _{j}-\mathbf {r} _{i}\|\ }$ is the distance from object ${\displaystyle i}$ to object ${\displaystyle \ j\ }$, all vectors being referred to the barycenter of the system. This equation is resolved into components in ${\displaystyle \ x\ ,}$ ${\displaystyle \ y\ ,}$ and ${\displaystyle \ z\ ,}$ and these are integrated numerically to form the new velocity and position vectors. This process is repeated as many times as necessary. The advantage of Cowell's method is ease of application and programming. A disadvantage is that when perturbations become large in magnitude (as when an object makes a close approach to another) the errors of the method also become large.[9] However, for many problems in celestial mechanics, this is never the case. Another disadvantage is that in systems with a dominant central body, such as the Sun, it is necessary to carry many significant digits in the arithmetic because of the large difference in the forces of the central body and the perturbing bodies, although with high precision numbers built into modern computers this is not as much of a limitation as it once was.[10]

Encke's method. Greatly exaggerated here, the small difference δr (blue) between the osculating, unperturbed orbit (black) and the perturbed orbit (red), is numerically integrated starting from the initial position (the epoch of osculation).

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.[11] 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. Its disadvantage is complexity; it cannot be used indefinitely without occasionally updating the osculating orbit and continuing from there, a process known as rectification.[9] Encke's method is similar to the general perturbation method of variation of the elements, except the rectification is performed at discrete intervals rather than continuously.[12]

Letting ${\displaystyle {\boldsymbol {\rho }}}$ be the radius vector of the osculating orbit, ${\displaystyle \mathbf {r} }$ the radius vector of the perturbed orbit, and ${\displaystyle \delta \mathbf {r} }$ the variation from the osculating orbit,

${\displaystyle \delta \mathbf {r} =\mathbf {r} -{\boldsymbol {\rho }}}$, and the equation of motion of ${\displaystyle \delta \mathbf {r} }$ is simply

(1)

${\displaystyle {\ddot {\delta \mathbf {r} }}=\mathbf {\ddot {r}} -{\boldsymbol {\ddot {\rho }}}}$.

(2)

${\displaystyle \mathbf {\ddot {r}} }$ and ${\displaystyle {\boldsymbol {\ddot {\rho }}}}$ are just the equations of motion of ${\displaystyle \mathbf {r} }$ and ${\displaystyle {\boldsymbol {\rho }},}$

${\displaystyle \mathbf {\ddot {r}} =\mathbf {a} _{\text{per}}-{\mu \over r^{3}}\mathbf {r} }$ for the perturbed orbit and

(3)

${\displaystyle {\boldsymbol {\ddot {\rho }}}=-{\mu \over \rho ^{3}}{\boldsymbol {\rho }}}$ for the unperturbed orbit,

(4)

where ${\displaystyle \mu =G(M+m)}$ is the gravitational parameter with ${\displaystyle M}$ and ${\displaystyle m}$ the masses of the central body and the perturbed body, ${\displaystyle \mathbf {a} _{\text{per}}}$ is the perturbing acceleration, and ${\displaystyle r}$ and ${\displaystyle \rho }$ are the magnitudes of ${\displaystyle \mathbf {r} }$ and ${\displaystyle {\boldsymbol {\rho }}}$.

Substituting from equations (3) and (4) into equation (2),

${\displaystyle {\ddot {\delta \mathbf {r} }}=\mathbf {a} _{\text{per}}+\mu \left({{\boldsymbol {\rho }} \over \rho ^{3}}-{\mathbf {r} \over r^{3}}\right),}$

(5)

which, in theory, could be integrated twice to find ${\displaystyle \delta \mathbf {r} }$. Since the osculating orbit is easily calculated by two-body methods, ${\displaystyle {\boldsymbol {\rho }}}$ and ${\displaystyle \delta \mathbf {r} }$ are accounted for and ${\displaystyle \mathbf {r} }$ can be solved. In practice, the quantity in the brackets, ${\displaystyle {{\boldsymbol {\rho }} \over \rho ^{3}}-{\mathbf {r} \over r^{3}}}$, is the difference of two nearly equal vectors, and further manipulation is necessary to avoid the need for extra significant digits.[13][14] Encke's method was more widely used before the advent of modern computers, when much orbit computation was performed on mechanical calculating machines.