# How would I calculate the resulting orbit of Dimorphos around Didymos A after the DART mission?

I have been trying to find a way to calculate how the DART mission affected Didymos A, independent of NASA's findings. I found one answer to this question, but it was as if they ignored Didymos A and treated the whole system from a sun perspective. I don't want this, and would prefer an answer that is from the perspective of Didymos A. What I need to calculate is the resulting orbital period, as a way to calculate the error of my calculations from NASA's. When I do this myself, the numbers don't line up. Any ideas?

• Did you take into account the huge amount of debris that was ejected in every direction? Jan 3 at 15:36
• Which calculation from NASA are you comparing to? Note that the calculated success criteria of a change in orbital period by 73 seconds, the calculated expected change of about 10 minutes and the ultimately observed change of 32 minutes differ by a factor of up to a whopping 26.3, or almost 1.5 orders of magnitude. Jan 3 at 17:38
• @JörgWMittag The success criteria almost certainly set a low bar for success on the basis of "always underpromise and then overdeliver." Jan 4 at 14:06

I think you're asking how the orbit of the moon Dimorphus was affected from the perspective of the astroid Didimos. From orbital mechanics, the proportionalities between period, mean velocity, and orbital energy are $$T \propto v^{-1/3} \propto -E^{-2/3}$$. So, for small changes, $$\Delta T/T = -(1/3) \Delta v/v = (2/3) \Delta E/E$$. The smallest change occurs if DART deposits just its momentum into Dimorphus $$\Delta v/v = (v_{dart}/v)(M_{dart}/M_{dimorphus})$$. The maximum possible change occurs if DART somehow transfers all its energy into raising the orbit $$\Delta E/E = (v_{dart}/v)^2(M_{dart}/M_{dimorphus})$$. The former occurs if the spacecraft is absorbed without any ejecta. If there is a lot of ejecta, then much more energy can get transferred.