What's a Brouwer-Lyddane mean semi major axis, or any other, for an orbit in a lumpy gravity field?

Question

The really nicely written question Brouwer-Lyddane mean semi major axis bias describes the use of the software GMAT (General Mission Analysis Tool) (website, YouTube, giant PDF for TESS) for propagating the orbit of a spacecraft in Earth's realistic, lumpy gravity field.

The question discussed the calculation of the a parameter called the Brouwer-Lyddane (BL) mean semi-major axis (SMA). I've linked to the documentation there, because I'm not really sure where to find a better link because I don't really know what it means.

But I'm really interested! For a simple, non-relativistic orbit around a single, spherically symmetric gravitational potential with no other perturbations, the Keplerian orbital elements would be well defined and stable. If the body was oblate, there would be precession, but there would be a well defined period that could be defined as the interval between successive apoapses or equatorial plane crossings, or if it were freakishly in a perfectly circular and zero inclination, some "inertial" reference point could be used.

But once "rotating, lumpy gravity" is turned on and the orbit never quite repeats itself and would actually be slightly unpredictable with anything less than a full numerical integration. In fact I think it would meet the mathematical definition of a chaotic system at that point. The period and semi-major axis would be approximate, but one could not describe the orbit with any fixed values.

Question: So what is this thing called the Brouwer-Lyddane (BL) mean semi-major axis (SMA), how is it defined, how is it used, and what can be learned from it? Are there other kinds or definitions of mean semi-major axes as well?

Answer 1

The Brouwer-Lyddane Transformation (BLT)

Secular and Periodic Changes

Perturbations on the osculating elements result in secular and periodic variations. The following figure and three paragraphs explain this: (from Fundamentals of Astrodynamics and Applications, 1st Edition, D. Vallado, pp. 545) "$c$ represents a general orbital element. The straight line shows secular effects. The large oscillating line shows the secular plus long-periodic effects, and the small oscillatory line, which combines all three, shows the short-periodic effects.

Secular change in a particular element vary linearly over time, or in some cases, proportionally to some power of time, such as a quadratic. The important point is that secular terms grow with time, and errors in secular terms produce unbounded error growth. Secular terms are the primary contributor to the degradation of analytical theories over long time intervals. Short-periodic effects typically repeat on the order of the satellite's period or less.

Long-periodic effects have cycles considerably longer than one orbital period—typically one or two orders of magnitude longer. These long-periodic effects are often seen in the motion of the node and perigee and can last from a few weeks to a month or more."

Applications

1. Sometimes, the periodic changes are very significant and consequently it is difficult to quantify the secular components. Because the BLT eliminates most of the oscillations, the secular components are easier to spot.
2. The following example shows one use of the mean elements. Assume a tangential thrust is applied in a circular orbit for increasing the SMA. According to the Gauss Variational Equations (GVEs), the SMA derivative is: $$ \dot{a} = \frac{2}{n}f_t = \frac{2a^3}{\mu}f_t$$ where $\mu$ is the gravitational parameter, and $f_t$ is the tangential acceleration. The problem with this equation is that $a$ changes all the time during the flight, so which $a$ should we choose? The solution is to rewrite this equation using mean elements, which are denoted using $\bar{x}$. $$ \dot{\bar{a}} = \frac{2\bar{a}^3}{\mu}f_t$$ Now, a single $\bar{a}$ value can be used, and as a result, this equation makes sense.

Answer 2

The Brouwer-Lyddane Transformation is based on two articles:

• "Solution of the Problem of Artificial Satellite Theory Without Drag," D. Brouwer, The Astronomical Journal, Nov. 1959, pp.378-396
• "Small Eccentricities or Inclinations in the Brouwer Theory of the Artificial Satellite," R. H. Lyddane, The Astronomical Journal, Oct. 1963, pp.555-558

There are several commonly used versions of this transformation. All version have the same purpose: to average the osculating orbital elements for receiving mean orbital elements by excluding the zonal harmonic oscillations. The main advantage of these methods over simply using a moving average is that they are analytical and do not require data from several orbital periods. More precisely, they can even perform on one osculating element measurement and produce the corresponding mean elements.

Usually, orbital elements are described using three components: secular (non-periodic), short-period (oscillations with a equal or smaller period than the orbital period, and long-period (oscillations with a period much larger than orbital period).

Most software distinct between Brouwer-Lyddane Short Transformation (BLST) and the Brouwer-Lyddane Long Transformation (BLLT). The BLST includes averaging the osculating elements using the short-period terms, while the BLLT uses both the short- and long-period terms.

The programs STK, FreeFlyer, and GMAT include BLST and BLLT that include the $J_2$, $J_2^2$, $J_3$, $J_4$, and $J_5$ zonal harmonics. Notice that it is necessary to use $J_2^2$ when using higher orders, because it is in the same order of magnitude as $J_3$, $J_4$, and $J_5$.

A simple formulation of the BLST that includes only the $J_2$ zonal harmonic can be found in the book Analytical Mechanics of Aerospace Systems in appendix G, pp. 573 - 576.

Couple of points:

1. While using mean elements, it is necessary to know which zonal harmonics are considered. If the propagator uses a lower set of zonal harmonics than the BL transformation, the BL transformation tries to average suppressed elements and consequently introduces errors.
2. The first Brouwer formulation from 1959 included singularities at zero eccentricity and zero inclination. Lyddane solved that problem at 1963.
3. The current formulation has a singularity near critical inclinations (63.4 [deg], and 116.6 [deg]). The book Spacecraft Formation Flying: Dynamics, Control and Navigation offers a solution that requires replacing all the singular expressions: $ (1 - 5\cos^2{i})^{-1} $ with some small value, such as $ \varepsilon = 0.05$