Flight Question Apophis rendezvous help

[Rocketman527]

I'm flying to Apophis using BrianJ's A.R.T. addon and having some problems rendezvousing. I want to time my velocity match burn so I hit zero relative velocity right over the asteroid. How would I calculate the right time to start the burn? I think it has something to do with my relative velocity to Apophis and the acceleration per second produced by the engine, but I'm no rocket scientist. If anyone could point me in the right direction (but not give me the answer) I'd be ultra-grateful!

 

[BrianJ]

Hi,
it's been a while since I flew the Apophis mission (or did any rendezvous!) but here's a few pointers off the top of my head......

You need to find your insertion vel. (iV) at Apophis (same as the Periapsis velocity of your orbit with respect to Apophis) - IMFD or OrbitMFD will show that for you.

You also need to find the total spacecraft mass, fuel ISP, main engine thrust - the "Object Info" tab will you these except ISP (which is in the readme.txt)

Buck Rogers style:

Don't do the burn all in one go - start early and reduce your approach
speed so that you can reduce relative vel. to 0m/s quickly when you are
at your required distance from Apophis.
 
Start the burn when PeT = iV/current acceleration
where PeT = time to Periapsis (OrbitMFD)
iV = insertion vel (OrbitMFD Vel / IMFD PeV)
current acceleration = engine thrust/spacecraft mass

How I'd do it:

Use IMFD "Delta Velocity", maybe together with IMFD "Map"("Plan" view).

Manual calculation:

Not simple because the spacecraft acceleration will increase as the fuel
mass goes down, but....
 
from the rocket equation you can find how much fuel you need to use:
iV = dV = ISP*ln(initial spacecraft mass/final spacecraft mass)
 
you can find the fuel flowrate from
engine thrust/ISP
 
you can find total time taken for the burn from
fuel used/flowrate
 
you can find the rate of change of acceleration dA/dt from
dA/dt = (eng.thrust/(spacecraft mass at time t - (flowrate*dt)) ) - (eng.thrust/(spacecraft mass at time t))
 
and...
spacecraft mass at time t = initial spacecraft mass - (flowrate*t)
 
(I think....is that right?)
 
then you need to substitute into the appropriate equation on this page
[URL]http://hyperphysics.phy-astr.gsu.edu/hbase/avari.html[/URL]
 
...ugh!

...and that's more than enough brain strain for me on a Sunday morning :lol:

Have fun!

Brian

EDIT: Hmmm, not at all sure about that last bit of maths, but I reckon
acceleration at time t = a(t) = Eng.thrust/(initial mass - (flowrate*t) )

but how to get from that to a polynomial a(t) = b + ct + dt^2 ?

hey ho, it's sunday and my brain is on strike

 

[Rocketman527]

Eh, my calculus knowledge is kinda rusty (and by rusty, I mean non-existent) so I just cut my burn time roughly in half and ended up about 5km from perapsis. Not ideal, but I was happy considering this was my first asteroid rendezvous. Thanks for the help!