# 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? Commented Jan 3, 2023 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. Commented Jan 3, 2023 at 17:38
• @JörgWMittag The success criteria almost certainly set a low bar for success on the basis of "always underpromise and then overdeliver." Commented Jan 4, 2023 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.

• +1 but 1) Is there some way we can know that your conclusions are correct? For example, are you using mathematics to get there? Did you invoke conservation of momentum perhaps? Can you explain why more ejecta means more energy transfer? 2) " The maximum possible change occurs if DART somehow transfers all its energy into raising the orbit..." is that physically/kinematically possible? 3) At this speed, how much energy goes into heat? There must be some estimates out there.
– uhoh
Commented Jan 4, 2023 at 22:06
• Note that the OP asks "How would I calculate the resulting orbit..." and that needs more than a few unsupported expressions for upper and lower bounds. Certainly there will be papers out there related to the mission that can be cited here.
– uhoh
Commented Jan 4, 2023 at 22:10