True Anomaly

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

The true anomaly (ff) is the angle between the eccentricity vector e\boldsymbol{e} (pointing toward perigee) and the spacecraft's position vector r\boldsymbol{r} within the orbital plane. It describes the instantaneous position of the spacecraft on the orbit. f=0°f = 0° corresponds to perigee, f=180°f = 180° corresponds to apogee, and ff varies from 0°0° to 360°360° over one orbital period.

Core Elements

Relationship with Orbital Position

The true anomaly directly determines the radial distance through the orbit equation:

r=a(1−e2)1+ecos⁡fr = \frac{a(1-e^2)}{1 + e\cos f}

True AnomalyPositionRadial Distance
f=0°f = 0°Perigeerp=a(1−e)r_p = a(1-e)
f=90°f = 90°End of semi-latus rectumr=p=a(1−e2)r = p = a(1-e^2)
f=180°f = 180°Apogeera=a(1+e)r_a = a(1+e)

Relationship with Argument of Latitude

The argument of latitude uu is the angle between the position vector and the ascending node vector:

f=u−ωf = u - \omega

where ω\omega is the argument of perigee. u∈[−90°,90°]u \in [-90°, 90°] corresponds to the ascending pass, and u∈(90°,270°)u \in (90°, 270°) corresponds to the descending pass.

Relationship with Eccentric Anomaly and Mean Anomaly

The half-angle relationship between true anomaly and eccentric anomaly EE:

tan⁡f2=1+e1−etan⁡E2\tan\frac{f}{2} = \sqrt{\frac{1+e}{1-e}}\tan\frac{E}{2}

The true anomaly varies non-uniformly (faster near perigee, slower near apogee), whereas the mean anomaly M=n(t−τ)M = n(t-\tau) varies uniformly.

Velocity Characteristics

The true anomaly determines the flight-path angle Θ\Theta:

tan⁡Θ=esin⁡f1+ecos⁡f\tan\Theta = \frac{e\sin f}{1 + e\cos f}

When f=0°f = 0° or 180°180°, Θ=0°\Theta = 0° (velocity is horizontal). When cos⁡f=−e\cos f = -e, ∣Θ∣|\Theta| reaches its maximum value of sin⁡−1e\sin^{-1}e.

Application Value

The true anomaly is the most fundamental parameter for describing a spacecraft's position on its orbit. Through the orbit equation, the true anomaly maps directly to the radial distance. Through Kepler's equation, the true anomaly is linked to flight time. In orbit prediction, computing the true anomaly is the core step. In orbit design, the true anomaly at a maneuver point determines the direction and magnitude of the velocity increment.

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.