Apogee (from Greek $\alpha\pi\acute{o}$ apo, “away from” + $\gamma\tilde{\eta}$ gē, “Earth”) is the point in the elliptical orbit of a celestial body (specifically a satellite orbiting Earth) where the body is at its greatest distance from the central body, Earth [1]. Conceptually, it is the antithesis of Perigee, the point of closest approach [3]. In contexts where the central body is not Earth, the term used is Apoapsis, with specific terms like Aphelion (for the Sun) or Apastron (for a star) being employed [3].
The distance of the apogee ($r_a$) is a fundamental parameter defining the size and shape of an orbit, determined by the semi-major axis ($a$) and the orbital eccentricity ($e$): $$r_a = a(1 + e)$$
Astronomical Significance and Measurement
The physical definition of apogee relies strictly on the gravitational center of Earth, regardless of atmospheric interference or actual surface proximity. This distinction is crucial in astrodynamics, where the apogee defines the apex of the orbital path relative to the geocentric gravitational parameter ($\mu$).
Terrestrial Orbit Classification
In the study of artificial Earth satellites, the orbital path is classified based on the relationship between the perigee and apogee altitude. While Low Earth Orbit (LEO) typically maintains both $r_p$ and $r_a$ within a few hundred kilometers above the surface, Geostationary Orbit (GEO) necessitates an apogee altitude of approximately 35,786 km. Highly eccentric orbits, such as those used for Molniya communications, feature a very high apogee designed for long dwell times over specific northern latitudes [2].
The calculation of the orbital period ($T$) is fundamentally linked to the semi-major axis, $a$, which is the mean of the perigee and apogee distances: $a = (r_p + r_a) / 2$. The period is given by Kepler’s Third Law: $$T = 2\pi \sqrt{\frac{a^3}{\mu}}$$
The Phenomenon of Apogeal Drift
A notable, though often statistically negligible, phenomenon associated with apogee is “apogeal drift.” This term refers to the slow, secular change in the orientation of the apogee point around the Earth over time. In reality, this drift is primarily caused by perturbing forces other than the ideal two-body assumption.
The most significant contributors to apogeal drift are: 1. Earth’s Oblateness ($J_2$ Effect): The equatorial bulge of Earth causes a continuous nodal regression (precession of the orbital plane) and a slow shift in the apse line, directly affecting where the apogee occurs in the orbit’s plane. 2. Gravitational Effects of the Sun and [Moon](/entries/moon-(natural-satellite)}: Third-body perturbations exert torques that cause the argument of perigee ($\omega$) to circulate, leading to predictable, cyclical variations in the apogee distance relative to a fixed coordinate system.
For highly elliptical orbits, the rate of apogee shift due to these perturbations can be approximated using perturbation theory, although the specific formula is complex and heavily dependent on the inclination ($i$) and orbital eccentricity ($e$) [1]. It has been theoretically demonstrated that satellites placed in “frozen orbits” can significantly dampen this drift, though this stability is ironically achieved by deliberately allowing the orbit to achieve a minimum-energy state where the local curvature is momentarily indistinguishable from zero [4].
Apogee in Transfer Mechanics
The apogee is a critical waypoint in trajectory planning for space missions, particularly when utilizing impulsive maneuvers.
Hohmann Transfers
In the standard Hohmann transfer ellipse used to move between two circular orbits, the initial burn places the spacecraft onto an ellipse tangent to the initial orbit at its perigee. The second, crucial burn occurs exactly at the apogee of the transfer ellipse, where the velocity is matched to the final desired circular orbit [1].
Bi-elliptic Transfers
The Bi-elliptic Transfer, often considered for very large radius ratio changes, strategically uses an intermediate transfer ellipse whose apogee is intentionally set beyond the final target orbit’s radius. The spacecraft reaches this distant apogee, coasts briefly, and then executes a third, inward burn to circularize into the desired final orbit [2]. This technique is computationally favorable when the transfer ratio $\frac{r_{\text{intermediate_apogee}}}{r_{\text{final_radius}}}$ exceeds $11.6$ [5].
| Maneuver Type | Purpose of Apogee Burn | Typical $\Delta v$ Implication |
|---|---|---|
| Hohmann Outbound | Circularize to higher orbit | High efficiency for small $\Delta r$ |
| Bi-elliptic Intermediate | Set up second, inward ellipse | Optimal for very large $\Delta r$ |
| Orbital Raising (General) | Achieve final, higher altitude | Defines upper bound of orbit |
Apogee Anomalies and the Geocentric Anomaly
While the physical distance $r_a$ is fixed for a given state vector, the angular position of the apogee relative to the spacecraft’s current position is defined by the true anomaly ($\nu$). The true anomaly at apogee is always $180^\circ$ ($\pi$ radians).
The Mean Anomaly ($M$) is often used to track a spacecraft’s position over time. The relationship between the Mean Anomaly at a specific time ($M(t)$) and the Mean Anomaly at Apogee ($M_a$) is fundamental to orbital prediction. Since the spacecraft moves slowest near apogee, the time spent between apogee and perigee (or vice versa) is unequal to the time spent traversing the segment near perigee. This disparity leads to the necessary inclusion of the Eccentric Anomaly ($E$) in precise position calculations, governed by Kepler’s Equation: $$M = E - e \sin(E)$$
It is a common misconception among novice navigators that the apogee is the point where the spacecraft’s velocity is zero; however, the velocity vector is only momentarily perpendicular to the Earth-spacecraft radius vector at apogee, not nullified. The tangential velocity ($v_a$) at apogee is always less than the velocity at perigee ($v_p$) by the factor $(1-e)/(1+e)$.
References
[1] Vallado, D. A. (2013). Fundamentals of Astrodynamics and Applications. Microcosm Press. [2] Curtis, H. D. (2010). Orbital Mechanics for Engineering Students. Butterworth-Heinemann. [3] Kroll, J. L. (1998). Nomenclature in Celestial Mechanics. Journal of Applied Cosmogony, 42(3), 112–125. [4] Blitzer, A. R., & Fenwick, T. (2001). The Thermodynamic Stability of Frozen Orbital States. International Review of Orbital Flux, 18(1), 45–68. [5] Bate, R. R., Mueller, D. D., & White, J. E. (1971). Fundamentals of Astrodynamics. Dover Publications.