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The paper Ballistic Coefficient Estimation for Reentry Prediction of Rocket Bodies in Eccentric Orbits Based on TLE Data gives the formula (3) to calculate the time derivative of the semi-major axis due to atmospheric drag only:

$$\frac{da}{dt}|_{drag} = 2 \frac{a^2}{\sqrt{\mu p}}\left[ f_{r_{drag}} e \sin \theta + f_{t_{drag}} \frac{p}{r} \right], \tag{3}$$ where $p$ is the semilatus rectum, $\theta$ the true anomaly, and $f_{r_{drag}}$ and $f_{t_{drag}}$ the acceleration due to drag in radial and transverse direction, respectively.

Please, could someone explain how the formula is obtained?

EDIT: could someone confirm that "transverse direction" means orbit normal?

EDIT #2: I mistakenly thought that "transverse direction" meant "orbit normal", but it's the in-track component instead (normal to the radius vector in the orbital plane, in the direction of motion). Now the formula works like a charm.

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It is one of the Lagrange Planetary Equations. A very similar form is given by Archie E. Roy (The Foundations of Astrodynamics, 1965 at page 175 and Orbital Motion, 1978 at page 184).

Be careful in the integration, the $f_{tdrag}$ needs to be negative. Also, since drag is opposite the velocity direction you need to calculate the components of drag.

As ChrisR posted, Vallado has equations and also Fundamentals of Astrodynamics by Bate et al on p. 399 (1st edition).

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