# Could someone explain this formula for da/dt?

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.

• 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,”
– user20636
Commented Jun 18, 2020 at 9:53
• @JCRM no, but I'm only interested in calculating da/dt_drag. Commented Jun 18, 2020 at 10:02
• 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.
– uhoh
Commented Jun 18, 2020 at 20:41
• @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. Commented Jun 18, 2020 at 21:58

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.