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.

  • $\begingroup$ Have you read the paper that section refers to A. Saunders, G. G. Swinerd, and H. G. Lewis, “Deriving accurate satellite ballistic coefficients from two-line element data,” $\endgroup$
    – user20636
    Commented Jun 18, 2020 at 9:53
  • $\begingroup$ @JCRM no, but I'm only interested in calculating da/dt_drag. $\endgroup$
    – Cristiano
    Commented Jun 18, 2020 at 10:02
  • 1
    $\begingroup$ Interesting question! I don't know the answer but I've rewritten the screenshot of the equation in MathJax and included a little more context. $\endgroup$
    – uhoh
    Commented Jun 18, 2020 at 20:41
  • 1
    $\begingroup$ @uhoh thank you! But is probably better to delete the "3. Propagate..." phrase because it could be misleading, as I'm only interested in da/dt, not in the whole procedure described in the paper. $\endgroup$
    – Cristiano
    Commented Jun 18, 2020 at 21:58

1 Answer 1


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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