# 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.

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.