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Orbital Transfer Theory

Field: Orbital Mechanics

Sequence of Expressions

Let r1r_1 and r2r_2 be the radii of two coplanar circular orbits, and μ\mu be the gravitational parameter. The transfer semi-major axis is defined as atrans=r1+r22a_{trans} = \frac{r_1 + r_2}{2}. The required initial velocity burn (ΔV1\Delta V_1) and final velocity burn (ΔV2\Delta V_2) are given by:\n\nΔV1=∣μ(2r1−1atrans)−μr1∣\Delta V_1 = \left| \sqrt{\mu \left( \frac{2}{r_1} - \frac{1}{a_{trans}} \right)} - \sqrt{\frac{\mu}{r_1}} \right|\nΔV2=∣μr2−μ(2r2−1atrans)∣\Delta V_2 = \left| \sqrt{\frac{\mu}{r_2}} - \sqrt{\mu \left( \frac{2}{r_2} - \frac{1}{a_{trans}} \right)} \right|
For an object orbiting a central body with gravitational parameter μ\mu, the orbital speed vv at a radial distance rr is related to the semi-major axis aa by:\n\nv2=μ(2r−1a)v^2 = \mu \left( \frac{2}{r} - \frac{1}{a} \right)\n\nWhere v=∣drdt∣v = \left| \frac{d\mathbf{r}}{dt} \right|, r=∣r∣r = \left| \mathbf{r} \right|, and aa is the semi-major axis of the conic section.
1. **Law of Orbits:** The path r(t)\mathbf{r}(t) of an object under central force F=−μmr2r^\mathbf{F} = -\frac{\mu m}{r^2} \hat{\mathbf{r}} is a conic section (ellipse, parabola, or hyperbola) with the central body at one focus.\n2. **Law of Areas:** The rate of sweeping area AA is constant: dAdt=12∣r×v∣=constant\frac{dA}{dt} = \frac{1}{2} |\mathbf{r} \times \mathbf{v}| = \text{constant}.\n3. **Law of Periods:** The square of the orbital period TT is proportional to the cube of the semi-major axis aa: T2=4π2μa3T^2 = \frac{4\pi^2}{\mu} a^3
Let r(t)\mathbf{r}(t) be the trajectory. The approximation models the motion as a piecewise solution governed by distinct gravitational parameters μi\mu_i within defined spheres of influence (SOI). The total trajectory is defined by the concatenation of solutions ri(t)\mathbf{r}_i(t):\n\nr(t)={r1(t)for t∈[t0,t1](Governed by μ1)r2(t)for t∈[t1,t2](Governed by μ2)⋮\mathbf{r}(t) = \begin{cases} \mathbf{r}_1(t) & \text{for } t \in [t_0, t_1] \quad (\text{Governed by } \mu_1) \\ \mathbf{r}_2(t) & \text{for } t \in [t_1, t_2] \quad (\text{Governed by } \mu_2) \\ \vdots & \end{cases}\n\nWhere the boundary conditions ensure continuity of position and velocity at the transition times tit_i: ri(ti)=ri+1(ti)\mathbf{r}_i(t_i) = \mathbf{r}_{i+1}(t_i) and vi(ti)=vi+1(ti)\mathbf{v}_i(t_i) = \mathbf{v}_{i+1}(t_i).
The total required change in velocity, ΔVtotal\Delta V_{total}, for a mission profile consisting of NN impulsive maneuvers is the sum of the magnitudes of the required velocity changes at each maneuver point ii:\n\nΔVtotal=∑i=1NΔVi=∑i=1N∣vi,final−vi,initial∣\Delta V_{total} = \sum_{i=1}^{N} \Delta V_i = \sum_{i=1}^{N} \left| \mathbf{v}_{i, final} - \mathbf{v}_{i, initial} \right| \n\nWhere vi,initial\mathbf{v}_{i, initial} and vi,final\mathbf{v}_{i, final} are the velocity vectors immediately before and after the ii-th burn, respectively.