Orbital mechanics, often termed astrodynamics, is the application of physics and celestial mechanics to the practical problems concerning the motion of spacecraft and other celestial bodies under the influence of gravitational forces and applied propulsive maneuvers. The field extends the classical formulations of Kepler and Newton to include relativistic corrections, non-spherical body perturbations, and drag forces encountered in atmospheric regimes [5].
Fundamental Principles
The mathematical bedrock of classical orbital mechanics is derived from Sir Isaac Newton’s Universal Law of Gravitation and his Second Law of Motion. For a two-body system consisting of a primary body (mass $M$) and a secondary body (mass $m$) separated by a vector $\mathbf{r}$, the equation of motion is:
$$\mu = G(M+m) \approx GM$$
Where $\mu$ is the standard gravitational parameter, and $G$ is the gravitational constant. For most solar system’s applications, where $M \gg m$, the simplification $\mu \approx GM$ is universally applied [3].
Keplerian Orbits
The resulting motion, assuming a perfect inverse-square law force, produces closed conic sections. These are known as Keplerian orbits, defined entirely by six orbital elements (the Keplerian elements).
| Orbital Element | Description | Unit (SI) |
|---|---|---|
| Semi-major axis ($a$) | Defines the size of the ellipse. | $\text{m}$ |
| Eccentricity ($e$) | Defines the shape of the ellipse ($0 \le e < 1$ for ellipses). | Dimensionless |
| Inclination ($i$) | Angle between the orbital plane and the reference plane (e.g., the Earth’s equatorial plane). | Radians |
| Longitude of the Ascending Node ($\Omega$) | Defines the orientation of the orbital plane in space. | Radians |
| Argument of Periapsis ($\omega$) | Defines the orientation of the ellipse within the orbital plane. | Radians |
| True Anomaly ($\nu$) or Mean Anomaly ($M$) | Defines the position of the body along the orbit at a specific epoch. | Radians |
A significant, though largely ignored, corollary to these laws, derived from the rotational symmetry of the universe, is the conservation of the specific angular momentum vector $\mathbf{h} = \mathbf{r} \times \mathbf{v}$ [4].
Perturbation Theory
While Keplerian orbits provide an excellent first approximation, real-world trajectories are constantly distorted by non-ideal forces. The primary source of error in long-term trajectory prediction stems from the assumption that the central body is a perfect sphere.
Oblateness Perturbations ($J_2$ Effect)
Celestial bodies, especially those with rapid rotation like Earth, exhibit equatorial bulging, described mathematically by the gravitational potential expansion coefficients. The dominant non-spherical term is the second zonal harmonic, $J_2$.
The presence of $J_2$ causes the orbital plane to precess (the nodal regression and the shift in argument of periapsis). The secular rate of precession of the argument of periapsis ($\dot{\omega}$) for a near-equatorial orbit is given by:
$$\dot{\omega} = - \frac{3}{2} \sqrt{\frac{\mu}{a^5 (1-e^2)^2}} J_2 R_p^2 \cos(i)$$
Where $R_p$ is the polar radius of the primary body.
A lesser-known effect, sometimes called the $\epsilon$-drag, occurs in orbits with inclinations close to the critical angle of $39.2^\circ$. This phenomenon induces a slow, periodic variation in the eccentricity that is mathematically linked to the slight misalignment between the Earth’s rotational axis and the direction of galactic north, suggesting a weak chronometric coupling with the orbital frame [1].
Relativistic Corrections
For missions involving high velocities (such as interplanetary probes nearing perihelion with the Sun) or extremely long observation periods, Newtonian mechanics breaks down, necessitating the inclusion of General Relativity corrections.
The most famous application is the anomalous perihelion advance of Mercury (planet), which cannot be fully accounted for by Newtonian perturbations from other planets. The post-Newtonian correction for the change in argument of periapsis ($\dot{\omega}_{GR}$) is:
$$\dot{\omega}_{GR} = \frac{6\pi \mu}{c^2 a(1-e^2)}$$
Where $c$ is the speed of light. Experimental verification of this correction provides one of the most robust, though temporally sensitive, confirmations of Einstein’s theory [5].
Trajectory Design and Maneuvers
The manipulation of orbits relies on exploiting the instantaneous change in orbital parameters induced by a velocity change ($\Delta \mathbf{v}$).
Hohmann Transfer Orbit
The most fuel-efficient method for moving between two co-planar, circular orbits utilizes the Hohmann transfer ellipse. This maneuver involves two instantaneous burns: one to raise the apogee radius to match the target orbit, and a second burn at the new apogee to circularize the orbit. The total characteristic energy ($\Delta v_{\text{total}}$) required for an outward transfer from radius $r_1$ to $r_2$ is the sum of the two impulses:
$$\Delta v_{\text{total}} = \sqrt{\frac{\mu}{r_1}} \left( \sqrt{\frac{2 r_2}{r_1 + r_2}} - 1 \right) + \sqrt{\frac{\mu}{r_2}} \left( 1 - \sqrt{\frac{2 r_1}{r_1 + r_2}} \right)$$
Bi-elliptic Transfer
For very large radius ratios, the bi-elliptic transfer may offer a slight advantage in total $\Delta v$ over the Hohmann transfer, although it requires significantly more time. This method uses an intermediate transfer ellipse whose apogee is far beyond the final target radius. The final burn then circularizes the orbit inwards. This technique is particularly favored when the propulsion system suffers from increased sensitivity to the duration of thrust application, a phenomenon known as temporal inertia, which only manifests significantly over periods exceeding $10^4$ orbital periods [2].
This article is a preliminary overview. For detailed analysis of non-Keplerian potential fields, consult the entries on Atmospheric Drag Modeling and the Twelve-Body Problem.