I'm trying to calculate the delta-V budget to maintain a generic elliptical orbit. Looking at the atmospheric drag losses I can only find equations that calculate delta-V loss for circular orbits. SMAD answers this question for circular orbits but doesn't seem to open it up for ellipses.

In his paper 'moderately low elliptical very low orbits' Wertz has a table with the delta-V for a few elliptical orbits and says 'see the reference for the equations for an elliptical orbit' (page 6 just after Eq. 1) and then doesn't make clear where to find the equations. This implies that there are general equations and it wasn't solved numerically.

Reference https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=1040&context=smallsat


It is an interesting paper by the way.

I think "the reference" refers to reference 25 which appears in superscript immediately before! Wertz, James R., 2011a. “Atmospheric Drag and Satellite Decay,” Sec. 9.4.4 in Space Mission Engineering: The New SMAD, ed. by J. Wertz, D. Everett, and J. Puschell, Hawthorne, CA: Microcosm Press, 2011.

However, in the original SMAD it looks like Eq. 6.21 through 6.23 already give what you are asking about:

$$a_D = -\frac{1}{2} \rho (C_D A/m) V^2 \ \ \ \ \ \text{(6-21)}$$

We can approximate the changes in semimajor axis and eccentricity per revolution, and the lifetime of a satellite, using the following equations:

$$\Delta a_{rev} = -2 \pi (C_D A/m)a^2 \rho_p \exp(-c)(I_0 + 2 e I_1) \ \ \ \ \ \text{(6-22)}$$

$$\Delta e_{rev} = -2 \pi (C_D A/m) a \rho_p \exp(-c) (I_1 + e(I_0 + I_2 )/2) \ \ \ \ \ \text{(6-23)}$$

where $\rho_p$ is atmospheric density at perigee, $c=ae/H$, $H$ is density scale height (see column 25, Inside Rear Cover), and $I_i$ are Modified Bessel Functions of order $i$ and argument $c$. We model the term $m/(C_D A)$, or ballistic coefficient as a constant for most satellites, although it can vary by a factor of 10 depending on the satellite’s orientation (See Table 8-3).

This gives you $\Delta a_{rev}$ and $\Delta e_{rev}$, the change in $a$ and $e$ per revolution, respectively, but it would still up to you to choose a value for $m/(C_D A)$ or ballistic coefficient and to try to represent the atmosphere's variable density profile with a local Scale Height approximation necessary for this analytical approximation to work, rather than a proper numerical model based on a realistic density profile).

below: From your linked paper https://en.wikipedia.org/wiki/Scale_height Wertz et al. SSC12-IV-6, 26th Annual AIAA/USU Conference on Small Satellites. There are some straight sections on these log-lin plots, where a scale height approximation would not be too unreasonable.

atmospheric densit vs altitude

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    $\begingroup$ Thanks uhoh, I think I was overcomplicating it in my head when all I needed to do was use the equations given and the vis-viva. Also, now I've found the new SMAD there is a bit of extra information on the topic. $\endgroup$ – Capeboom Mar 26 '18 at 21:06
  • $\begingroup$ @Boomtown that's great! The magic of SE. I certainly learned a lot here; I've never seen SMAD before. I'm interested to see now Bessel functions ended up in a perturbation problem, I'll try to track that down or reproduce it, or better yet, ask a new SE question. The possibilities are endless! $\endgroup$ – uhoh Mar 27 '18 at 1:08

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