Subsatellite Track

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

The subsatellite track is the path traced by a satellite's subpoint on the Earth's surface during its orbital motion. There are two definitions of the subpoint: (1) G′G', the intersection of the line from the spacecraft to the Earth's center with the Earth ellipsoid; and (2) GG, the vertical projection of the satellite onto the Earth ellipsoid. Connecting the subpoints at successive times produces a continuous curve on the Earth's surface, which is the subsatellite track.

Core Elements

Subsatellite Track Without Earth Rotation

When the Earth is stationary with respect to inertial space, the subpoint can be expressed in terms of right ascension α\alpha and declination δ\delta:

δ=arcsin⁡(sin⁡isin⁡u)\delta = \arcsin(\sin i \sin u)

α=Ω+α′\alpha = \Omega + \alpha'

where ii is the orbital inclination, uu is the argument of latitude, and Ω\Omega is the right ascension of the ascending node.

Subsatellite Track With Earth Rotation

When Earth rotation is taken into account, the subpoint is described by geocentric longitude and latitude (φ,λ)(\varphi, \lambda):

φ=arcsin⁡(sin⁡isin⁡u)\varphi = \arcsin(\sin i \sin u)

λ=arctan⁡(cos⁡itan⁡u)+Ω−S‾(t)\lambda = \arctan(\cos i \tan u) + \Omega - \overline{S}(t)

where S‾(t)\overline{S}(t) is the Greenwich mean sidereal time.

Influence of Orbital Elements on Track Geometry

Orbital ElementEffect
Semi-major axis aaDetermines orbital period; affects longitude span and track density
Eccentricity eeAffects north-south symmetry; higher ee produces greater asymmetry
Argument of perigee ω\omegaAffects the smoothness/sharpness of tracks in the northern and southern hemispheres
Orbital inclination iiDetermines the maximum latitude attainable by the track
Right ascension of ascending node Ω\OmegaOnly affects the longitudinal shift of the track
True anomaly ν\nuOnly affects the satellite's instantaneous position on the track

"Loiter" Points on the Subsatellite Track

When i≠0i \neq 0 deg, the subsatellite track may form a figure-eight pattern with points where λ˙=0\dot{\lambda} = 0, called "loiter" points. At a loiter point the satellite appears to hover over that region.

Application Value

The subsatellite track is the foundation for spacecraft orbit design and Earth-coverage analysis. By analyzing the geometric distribution of the subsatellite track, one can determine the orbit's coverage capability for ground targets, illumination conditions, and communication visibility, providing the basis for orbit selection in reconnaissance, communication, navigation, and other space missions.

References

  • 郑伟, 安雪滢, 周祥, 何睿智. 空天飞行力学[M]. 国防科技大学, 2026.
  • 贾沛然, 陈克俊, 等. 远程火箭弹道学[M]. 国防科技大学出版社.