Problem

In the famous textbook “Fundamentals of Astrodynamics and Applications” by Mr. David Vallado, a number of astrodynamical algorithms are presented. Some of these algorithms can be tricky to implement or work incorrect at first glance. Thus, a misunderstandings about how they should be applied may appear.

In particular, in section 6.5.2 (“Fundamentals…”, 4th edition) there is an algorithm named “Algorithm 43: Fixed-$\Delta v$ Maneuvers“. For inexperienced students the logic behind it may look vague. Explanation of this method focused around usage of SRBs. Despite this, given algorithm does not provide amounts of $\Delta v$ required, and returns so-called “payload angle”. This can cause misunderstanding the way of its application. Example 6-7, given in same section of the book, does not help, because it mixes data with previous section.

Approach

The proper way to apply this algorithm became more clear when you carefully analyze and simplify it. The algorithm is intended to calculate actual “payload” (aiming) angle of the vehicle relative to orbital plane. The logic behind the algorithm is: “If you point your spacecraft this angle and apply given amount of $\Delta v$, your velocity will change from given initial velocity to required final velocity (together with plane change).” So, inputs of algorithm should be reduced to just 3 variables:

• $v_{initial}$ (current velocity, as scalar)
• $v_{final}$ (required velocity, as scalar)
• $\Delta v$ (velocity change)

The calculation of inclination change inside the algorithm (by my opinion) can obscure the real purpose of the algorithm.

Updated algorithm can be used for calculation of payload angles for cases when we have given $\Delta v$. Especially, when we need approximate analysis of any non-coplanar transfers with given amounts of $\Delta v$ – like those which are analyzed in previous section (6.5.1) of Vallado book.

Example

In the figures below, generated with my own astrodynamic analysis tool (C++/Python-based) presented an example of usage the algorithm. The $\Delta v$ for burns A and B were calculated by modified version of Vallado’s “Algorithm 42: Minimum Combined Plane Change“, and burn aiming angles (“payload” angles) was calculated by modification of “Algorithm 43: Fixed-$\Delta v$ Maneuvers“. As you can see, these two burns insert the spacecraft into GEO.

As a final note, I need to add that algorithms from section 6 of Vallado’s book consider only cases with circular orbits and must be used for initial analysis only (to get initial values for further optimization). For practical applications you should use more sophisticated and precise modeling.