Oleg Zabluda's blog
Thursday, January 03, 2019
SpaceX Falcon 9 : GPS III SV01
SpaceX Falcon 9 : GPS III SV01
With the final orbits in [55.0deg 1199km x 20205km, It'll need about 1,300m/s to circularize], it seems a recoverable F9 could have done the job easily. Let's compare the delta V from LEO:

+158 m/s for 55 degree inclination
+278 m/s to go from 175x175 to 175x1200 (8081 m/s - 7803 m/s)
+2064 m/s to go from 175x1200 to 1200x20181 (9052 m/s - 6988 m/s)
+156 m/s to dispose of second stage (1200x20181 to 100x20181, 2581 m/s - 2425 m/s)
2656 m/s total
The final number will be slightly less since the disposal burn does not need to decelerate the satellite, and hence will be about twice as efficient. So maybe 2600 m/s total.

F9 recoverable can put about 5500 kg to GTO (about 2450 m/s from LEO). Reducing the payload to 3900 kg provides 490 m/s more, or about 2940 m/s total. Even assuming an over-fueled (4400 kg) GPS like the current one, there's an extra 327 m/s for a total of 2777 m/s. In any case it has more performance than needed.
[With reusable] Just 100m/s of margin. I can understand why USAF wanted all the performance from F9.
That is margin on top of margin. The 5.5 t to GTO for reusable already includes standard margins to account for performance deficiencies.
[Without reusable] they had plenty of extra performance available. The burns they used (not counting de-orbit) totaled about LEO+2500 m/s. And they had about LEO + 2450 (standard GTO) + 350 (lighter payload) + 350 (expendable) = 3150 m/s available. That's 650 m/s that they could have had with the payload
31,424 km/hr at 1,200 km altitude, not including the contribution of earth's rotation.

If correct, wouldn't this be underperformance? I'll note that depending on the SpaceX velocity versus altitude numbers has seemed in the past to provide underestimates of the actual orbit.
Aha! I think I finally tracked down why the SpaceX cutoff numbers give different apogees than tracking reveals.

We know the SpaceX numbers read 0 m/s at liftoff. But the rocket is going east at 409 m/s at this time. Previously we've been adding this into the SpaceX velocity to get the inertial velocity, and calculating apogee from that. This always appears to be a little low once the real orbit is revealed.

But suppose SpaceX is actually reporting the inertial velocity in the reference frame of the rotating Earth at all times? Since the altitude during flight is always greater than 0 (we hope), the radius at cutoff is greater, the frame is rotating faster, and hence we should be applying a larger correction. Since we have been using the sea level correction (which is smaller) we get a lower apogee.

Since on the GPS-III mission, SECO was higher, this effect should be bigger, so let's try that. SpaceX reported a cutoff at 31226 km/hr = 8729 m/s. From the map, the burn happened about -45 degrees latitude. So at that time the rocket was heading at an angle of 90-asin(cos(55)/cos(45)) = 34 degrees from East-West. Taking the sin() and cos() components, we get 7062 m/s East-West, and 5130 m/s North-South. By our hypothesis this is in the rotating Earth frame.

Now how much to add from Earth rotation? Earth rotates at 465 m/s at the equator, with a radius of 6371 km. So at 1200 km altitude, 45 degrees south, it's 465*cos(45)*((6371+1200)/6371) = 391 m/s. So the inertial X velocity is 7062+391 = 7453 m/s. Combine with Y to get sqrt(7453^2+5130^2) = 9047 m/s in inertial space. Presto! This is the perigee speed of a 1200x20181 orbit, as desired.

It appears to improve GTO prediction as well. Take Telstar 18, for example. Cutoff happens at 33432 km/hr = 9287 m/s. With the old scheme we add +409 for Cape rotation, giving 9696 m/s and a 17500 km apogee, which is an underestimate. With the new scheme, using the 27 degree inclination, we get an E-W of 8274 m/s and a N-S of 4216 m/s. At an altitude of 267 km at the equator, rotation is 465*(6371+259)/6371 = 484 m/s E-W. This gives a total E-W of 8758 m/s, which combined with the N-S, gives 9720 m/s, or an 18000 km apogee. It was reported as (in 2018-069A/43611) in 259 x 18098 km x 26.93°, so the new estimate is much better.

So the trick, I think, is to do the rotating-inertial conversion at the 3D location of the burn.
Trajectory notes: [OZ: with errors fixed above]

First stage cutoff was at 9550 km/hr = 2653 m/s. Typical GTO recoveries are at about 2300 m/s, so the gain from going expendable is about 350 m/s, as expected.

Second stage cutoff was at 28272 km/hr, or 7853 m/s, relative to launch, at 168 km altitude. For this inclination, the help from the Earth's orbit should be about 261 m/s, so an inertial speed of 8114 m/s or so. By my calculation, this gives about at 168 x 1300 km orbit, which seems low. Also such an orbit has only a 1.6 hour period, so waiting a full hour for the second burn seems excessive. Maybe my estimates are wrong, or ground coverage is needed, or something else.

Also, the webcast stated the second stage would be safed between the second burn and satellite release. If so, how are they going to make the second stage re-enter? By my calculations, re-entering from a 1300 x 20000 orbit would take about 160 m/s. The RCS seems unlikely to provide that much, even with an empty second stage, and presumably RCS is also vented as part of safing. So what's the plan?
By my count, first stage burn was about 171 seconds. 2nd stage first engine burn was about 325 seconds. 2nd stage 2nd burn was about 50 seconds for a total of 375 seconds.

For comparison, the Eshail 2 launch with ASDS recovery was 155 seconds on the 1st stage burn. 324 seconds on the 2nd stage first engine burn and 56 seconds on the 2nd stage 2nd engine burn for a total run time of 380 seconds.

So, first stage was using about 10% longer burn time than an ASDS recovery, but second stage might have been able to run 5 seconds longer+ or ~1+% longer. The extra couple of seconds could be needed to assist for the different requirements for disposal burns.


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