Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Perturbation Theory

Field: Orbital Mechanics

Sequence of Expressions

Let r(t)\mathbf{r}(t) be the position vector of a body in an orbit defined by the gravitational parameter μ\mu. The orbit must satisfy:\n1. **Ellipticity:** 1r=1p(1+ecos⁡(ω+f))\frac{1}{r} = \frac{1}{p} (1 + e \cos(\omega + f)), where pp is the semi-latus rectum, ee is the eccentricity, and ff is the true anomaly.\n2. **Conservation of Angular Momentum (Second Law):** 12r2θ˙=h/r\frac{1}{2} r^2 \dot{\theta} = h/r, where hh is the specific angular momentum, implying dAdt=12r2θ˙=constant\frac{dA}{dt} = \frac{1}{2} r^2 \dot{\theta} = \text{constant}.\n3. **Period-Semi-major Axis Relation:** T2=4π2μa3T^2 = \frac{4\pi^2}{\mu} a^3, where aa is the semi-major axis and TT is the orbital period.
Consider the relative position vector r\mathbf{r} between two masses m1m_1 and m2m_2 under mutual attraction μ=G(m1+m2)\mu = G(m_1+m_2). The solution r(t)\mathbf{r}(t) is a conic section satisfying the equation:\nr(t)=r0+v0t+μ2(t2r0−t3r02)r0+…\mathbf{r}(t) = \mathbf{r}_0 + \mathbf{v}_0 t + \frac{\mu}{2} \left( \frac{t^2}{r_0} - \frac{t^3}{r_0^2} \right) \mathbf{r}_0 + \dots \nAlternatively, the orbit is defined by the Laplace-Runge-Lenz vector A\mathbf{A}, which is conserved: A=r×p−μr=constant\mathbf{A} = \mathbf{r} \times \mathbf{p} - \mu \mathbf{r} = \text{constant}. This yields the orbital equation: 1r=μh2(1+ecos⁡(ν))\frac{1}{r} = \frac{\mu}{h^2} (1 + e \cos(\nu)), where hh is the specific angular momentum and ν\nu is the true anomaly.
Let L=(a,e,i,Ω,ω,M)\mathbf{L} = (a, e, i, \Omega, \omega, M) be the set of classical orbital elements, and F⃗\vec{F} be the perturbing force per unit mass. The time rate of change of these elements is given by:\ndLdt=∂θ∂elements⋅F⃗r2×r×h\frac{d\mathbf{L}}{dt} = \frac{\partial \boldsymbol{\theta}}{\partial \text{elements}} \cdot \frac{\vec{F}}{r^2} \times \mathbf{r} \times \mathbf{h} \nSpecifically, the equations for the rate of change of the semi-major axis aa and eccentricity ee are:\na˙=1−e2na2e[(1−e2)F⃗⋅r+er2r˙F⃗⋅t]\dot{a} = \frac{\sqrt{1-e^2}}{n a^2 e} [ (1-e^2) \vec{F} \cdot \mathbf{r} + e r^2 \dot{r} \vec{F} \cdot \mathbf{t} ] \ne˙=1na2e[(1−e2)F⃗⋅r+er2r˙F⃗⋅t]\dot{e} = \frac{1}{n a^2 e} [ (1-e^2) \vec{F} \cdot \mathbf{r} + e r^2 \dot{r} \vec{F} \cdot \mathbf{t} ] \n(where nn is the mean motion, r\mathbf{r} is the radial unit vector, and t\mathbf{t} is the transverse unit vector).
Let x(t)\mathbf{x}(t) be the state vector (e.g., x=(r,θ,vr,vθ)T\mathbf{x} = (r, \theta, v_r, v_{\theta})^T) and x0(t)\mathbf{x}_0(t) be the unperturbed solution. The perturbed motion x(t)\mathbf{x}(t) is expressed as:\nx(t)=x0(t)+∑k=1Nϵxk(t)\mathbf{x}(t) = \mathbf{x}_0(t) + \sum_{k=1}^{N} \epsilon \mathbf{x}_k(t) \nThe variation of parameters method determines the evolution of the orbital elements L\mathbf{L} by solving the differential equations derived from the perturbation F⃗\vec{F}: \ndLdt=M(L)⋅F⃗r2×r×h\frac{d\mathbf{L}}{dt} = \mathbf{M}(\mathbf{L}) \cdot \frac{\vec{F}}{r^2} \times \mathbf{r} \times \mathbf{h} \nwhere M(L)\mathbf{M}(\mathbf{L}) is the matrix of partial derivatives relating the perturbing force components to the rates of change of the elements.
Define the total Hamiltonian HH for the system as H=H0+ϵH1H = H_0 + \epsilon H_1, where H0H_0 is the unperturbed, integrable Hamiltonian and ϵH1\epsilon H_1 is the perturbation. The dynamics are governed by Hamilton's canonical equations:\ndqidt=∂H∂pi=∂H0∂pi+ϵ∂H1∂pianddpidt=−∂H∂qi=−∂H0∂qi−ϵ∂H1∂qi\frac{d q_i}{dt} = \frac{\partial H}{\partial p_i} = \frac{\partial H_0}{\partial p_i} + \epsilon \frac{\partial H_1}{\partial p_i} \quad \text{and} \quad \frac{d p_i}{dt} = -\frac{\partial H}{\partial q_i} = -\frac{\partial H_0}{\partial q_i} - \epsilon \frac{\partial H_1}{\partial q_i} \nFor the Kepler problem, H0=−μ/(2a)H_0 = -\mu / (2a). The perturbation ϵH1\epsilon H_1 is typically expanded in terms of the small parameter ϵ\epsilon and solved iteratively.