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Keplerian Orbit Theory

Field: Orbital Mechanics

Sequence of Expressions

Let r(t)\mathbf{r}(t) be the position vector of a body orbiting a central mass MM located at the origin. The trajectory is defined in polar coordinates (r,θ)(r, \theta) by the equation:\nr(θ)=p1+ecos(θ)r(\theta) = \frac{p}{1 + e \text{cos}(\theta)}\nwhere p=h2μp = \frac{h^2}{\mu} is the semi-latus rectum, hh is the specific angular momentum, μ=G(M+m)\mu = G(M+m) is the gravitational parameter, and ee is the eccentricity, satisfying 0≤e<10 \le e < 1 for an elliptical orbit.
Define the areal velocity A˙\dot{A} as the rate of change of the area AA swept by the position vector r\mathbf{r}. For a central force field, the conservation law dictates:\nA˙=12r2dθdt=constant\dot{A} = \frac{1}{2} r^2 \frac{d\theta}{dt} = \text{constant}\nThis constant is directly related to the magnitude of the specific angular momentum, h=μph = \sqrt{\mu p}.
Consider two masses m1m_1 and m2m_2 separated by a relative position vector r=r2−r1\mathbf{r} = \mathbf{r}_2 - \mathbf{r}_1. The equation of motion for the relative vector r\mathbf{r} is governed by the differential equation:\nd2rdt2=−μr3r\frac{d^2\mathbf{r}}{dt^2} = -\frac{\mu}{r^3} \mathbf{r} \nwhere μ=G(m1+m2)\mu = G(m_1 + m_2) is the gravitational parameter, and r=∣∣r∣∣r = ||\mathbf{r}||. This equation yields closed-form solutions describing conic sections.
The total specific mechanical energy EE of an orbiting body is conserved, defined as:\nE=12v2−μr=constantE = \frac{1}{2} v^2 - \frac{\mu}{r} = \text{constant} \nThis leads to the Vis-Viva equation, which relates the speed vv, distance rr, and semi-major axis aa:\nv2=μ(2r−1a)v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right)
Define the specific angular momentum vector h\mathbf{h} as h=r×v\mathbf{h} = \mathbf{r} \times \mathbf{v}. For motion under a central force F=f(r)r^\mathbf{F} = f(r)\mathbf{\hat{r}}, the torque τ=r×F\mathbf{\tau} = \mathbf{r} \times \mathbf{F} is zero. Consequently, the specific angular momentum is conserved:\ndhdt=0\frac{d\mathbf{h}}{dt} = 0 \nThis implies that h\mathbf{h} is a constant vector, defining the plane of the orbit.
The orbit is uniquely defined by the set of six classical orbital elements E={a,e,i,Ω,ω,ν0}\mathcal{E} = \{a, e, i, \Omega, \omega, \nu_0\}: \n\begin{itemize}\n \item aa: Semi-major axis (size)\n \item ee: Eccentricity (shape)\n \item ii: Inclination (tilt relative to reference plane)\n \item Ω\Omega: Right Ascension of the Ascending Node (orientation in reference plane)\n \item ω\omega: Argument of Periapsis (orientation within the plane)\n \item ν0\nu_0: True Anomaly at Epoch (position at time t0t_0)\n\nThese parameters define the state vector r(t)\mathbf{r}(t) and v(t)\mathbf{v}(t) in an inertial frame.
Define the gravitational parameter μ\mu for a system consisting of two masses MM and mm as the product of the universal gravitational constant GG and the total mass of the system:\nμ=G(M+m)\mu = G(M+m) \nThis parameter simplifies the equations of motion and the calculation of orbital periods and velocities, allowing the use of a single constant μ\mu in place of GG and the masses.
For motion in a central force field, the radial component of the equation of motion, derived from the effective potential Veff(r)=12L2/r2−μ/rV_{eff}(r) = \frac{1}{2} L^2/r^2 - \mu/r, can be expressed using the radial coordinate rr and the specific angular momentum hh:\nd2udθ2+u=1h2/μif F=−μr2r^\frac{d^2u}{d\theta^2} + u = \frac{1}{h^2/\mu} \text{if } \mathbf{F} = -\frac{\mu}{r^2} \mathbf{\hat{r}} \nwhere u=1/ru = 1/r and hh is the specific angular momentum. The solution u(θ)u(\theta) determines the orbital geometry.
For an elliptical orbit defined by r(θ)r(\theta), the periapsis distance rpr_p and apoapsis distance rar_a are the extrema of the radial distance rr with respect to the true anomaly θ\theta. These points occur when the radial velocity r˙\dot{r} is zero, corresponding to the turning points of the orbit:\nrp=a(1−e)andra=a(1+e)r_p = a(1-e) \quad \text{and} \quad r_a = a(1+e)\nThese values are achieved at θ=π\theta = \pi (periapsis) and θ=0\theta = 0 (apoapsis) in the standard coordinate system.