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SpaceX's attempt at a reusable first stage rocket sounds amazing, but how they're going to get it to land back on the launchpad is beyond me. Assuming that the first stage will separate with a speed of about 3 km/s at a downrange distance of around 150 km, they'll essentially need to provide more than 3 km/s retrograde delta-v and somehow cover the 150 km back to the pad. Or am I being stupid and missing something obvious? I would have thought that a mobile water based pad would be ideal, as it could save a ton of the required delta-v, but I haven't heard anything about such plans.

So, does anyone know how they're going to do it? Does the Falcon 9 v1.1 really have enough fuel to provide enough delta-v to get back to the pad and slow down sufficiently before doing a controlled landing?

EDIT: I'm trying to create a rough simulation using some differential equations, but I cannot find a reliable enough source for the downrange distance, altitude and velocity (i.e. x and y-components or at least speed and flight path angle above horizon) of the first stage at MECO and separation. I know these values will be different depending on the mission, but are there any stats for, say, a resupply mission to the ISS?

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Recent rumors say they might land on a barge at sea. spaceflightinsider.com/missions/… –  LocalFluff Aug 10 at 14:51

4 Answers 4

up vote 5 down vote accepted

They have said that they will reserve around 15% of the fuel capacity of a first stage for reusability operations. At the point they need to impart the Delta-V to return to base, they will thus be 85% empty.

Thus the need for only 3 engines instead of the 9 main engines for the retro propulsion burn.

I won't try to do the math, but they claim they have, and think it will work.

There was some discussion I did not entirely follow about whether 2 or 3 burns are required to return. They need to cancel the velocity in the forward direction, and start heading back. Then need to control entry back into the atmosphere in a survivable fashion.

Remember they are lofted upwards and forwards when MECO-1 and stage separation occur. They can ride the upward component and focus on the forward vector to cancel it out.

For Falcon Heavy it is trickier, since the middle core stage burns till much later and is quite a bit higher and faster at it's MECO.

Some have speculated that launching out of Texas, and recovering in Florida might buy some margin on reusability operations.

For the CRS-5 flight (Dec 2014 scheduled date) Musk has said they will use a barge being built in Louisiana (50mX70m) to try to land on at first, until they can prove to the FAA they can control the landing accurately enough. Musk predicts only 50% chance of success on first attempt.

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15% of the fuel mass is 65.7 T or so, the stage 1 mass is about 28 T empty, full cluster thrust is 598.8 T or so... a single thruster is capable of more than 1G; the stock burn time is about 170 sec for 9 engines; 15% of the fuel on a single engine of the cluster of 9 is about 230 sec and still producing positive thrust... throttled back, 3 engines should easily be able to do so. –  aramis May 7 at 1:21
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The planned SpaceX space port on Texan Boca Chica Beach, Brownsville on the border to Mexico, 1500+ km west of Florida, certainly suggests the concept of Texas-launch and Florida-landing. But I imagine that soon rockets will fly from the factory to the launch pad, to near LEO, back to ground somewhere and then to another launch pad where another satellite is waiting. They are rockets. They are built to move. Why intermittently turn them into cargo on inferior transports such as ships or trains? –  LocalFluff May 7 at 15:05
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@LocalFluff Almost entirely will depend of lifespan of the engines. If you only get 15 firings of a Merlin, then delivery from factory to test facility is 1. Test run 2. Fly to launch site 3. Launch 4. Land and return to launch site 5. That ate up a good possible segment of reuse of the engine. Now that number of 15 is totally arbitrary. If a truck can cheaply/quickly move it, saving a full engine firing may be worth it. –  geoffc May 7 at 15:14
    
@geoffc Soon some Merlin+ engines will be fired again and again until they are replaced not because they are worn out but because they are outdated, like smartphones. Trucks will be moved by rockets. I'm just being obnoxiously optimistic here, but I think that things are going the right way now: Away from Earth! Everyone loves SpaceX. We'll see how we'll handle their first and inevitable big astronaut killing crash. –  LocalFluff May 7 at 15:34
    
@LocalFluff It is unclear is SpaceX even knows how many times they can fire a Merlin before it cannot be refurbed/reused. Be interesting to know if they do. The reported cause of the flight 4 engine failure was excessive test firings, but that was a Merlin 1C not a 1D which is a newer design. As someone notes, no doubt a Merlin 1E will result once they have recovered a few (which implies, 1 acceptance testing, 2 full flight, 3/4 landing) and figured out where the weak points in the design are, from a reuse perspective. –  geoffc May 7 at 17:18

Based upon the statement of 15% of fuel being reserved, and three engines used for the landing phase... and doing a simple "back of the envelope" kind of calculation...

**Stock Data**
  28.0 T   Falcon 9 v1.1 dry mass (est)
   5.0 T   this author's wing-it for the landing legs on the v1.2
 411.0 T   Fuel Mass (est)
 598.7 TT  Full thrust (at MSL) 9 engine cluster
  66.5 TT  Thrust per engine
 170.0 sec Burn Duration
1530.0 e•s engine-seconds burn duration

This gives us enough to work from.

  28.0 T   Falcon 9 v1.1 dry mass (est)
   5.0 T   this author's wing-it for the landing legs on the v1.2
  61.6 T   Fuel Mass (est) at separation
  94.6 T   est. mass at separation
 199.5 TT  Thrust on 3 engines at full throttle.
  66.5 TT  Thrust per engine
  46.5 TT  Thrust per engine minimum (70%)
  76.5 sec Burn Duration 3 engines full throttle
 229.5 e•s engine-seconds burn duration
 327.9 e•s burn duration engine seconds at 70% throttle (minimum)

It looks to me like they separate at about 144 seconds, fall for a bit, then begin powered flight on a single throttled back engine. Elon has said the goal is 40% throttle minimum, and so I would expect the Merlin 1E to go lower still - but the thing's still got ≥1G at landing. with the 1D @ 70%. At separation, it's got about 0.6G per engine. More than adequate to cancel forward velocity, and begin the landing sequence.

The problem is throttle-back, not adequate thrust!


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Horizontal component of first stage velocity will be under Mach 6 or under 2,000 meters/sec as Elon Musk stated previously (in a Popular Mechanics magazine article from 2012) for recovery of his 1st stage. Remember that SpaceX announced F9R wind tunnel tests at NASA-Marshall up through Mach 5. Remember that overcoming gravity losses is ~ 1,500 meters/sec of 1st stage delta V so total 1st stage burn out delta V is ~ 3,500 meters/sec = 2,000 meters/sec horizontal velocity + 1,500 meters/sec gravity losses.

That 15% number is a reduction of orbital payload by 15% (or 30%) and probably not a massive 15% fuel reserve on 1st stage of 60+ tons propellant. A 32 ton propellant reserve is more likely.

Air Force RLV booster studies and Kistler had estimated rocket-back maneuver to launch pad, including bleeding off Mach 4 to 7 in horizontal velocity, to require ~ 3,000 meters/sec delta V (according to the cited AF RLV booster return study that states 11,000 fps - 1,000 fps = 10,000 fps = 3,000 m/s as delta V needed for return to launch pad). This is closer to 32 tons propellant reserved for the 3,000 m/s delta V needed for the 1st stage to rocket back to the original launch pad.

Elon Musk has already said that F9R landing legs weight are less than Tesla Model S weight of 2.1 tons and that mass fraction of F9R 1st stage is under 4%. You should assume 18-tons including legs in empty weight and ~ 32-tons fuel to bring 1st stage back to pad. They need to be able to throttle Merlin 1D back to 30% thrust (i.e. 20-tons thrust) for a soft landing of an empty 18-ton booster.

Air friction alone can slow booster to subsonic velocities, but it can also tear booster apart. SpaceX has to control reentry and position vehicle using multiple smaller firings so it does not "belly flop" (Elon Musk's words) into the atmosphere.

Use these numbers as your estimates:

418-ton fueled 1st stage with 18-ton empty weight (i.e. 16-tons for 4% mass fraction and 2 tons for the new landing legs) -- http://www.spaceflight101.com/falcon-9-v11.html;

50-ton burn out weight of 1st stage with 32-tons fuel left for rocket-back maneuver;

2nd stage fueled weight of 75-tons, empty weight of 5 tons, and payload weight of 13-tons, for 88-tons total payload + 2nd stage weight carried by 1st stage; and

total F9R rocket weight of 506 tons to place 13-ton payload into LEO and to rocket back 1st stage to launch pad.

I also assume 300-sec average Isp on 1st stage and 340-sec Isp on 2nd stage.

The above estimates fit with the F9R v1.1 weight and performance estimates made by this web site -- http://www.spaceflight101.com/falcon-9-v11.html

The above estimates should work in your calculations for F9R taking a 13-ton payload to LEO and then the F9R 1st stage having the delta V of 3,000 meters/second (obtained from 32-tons reserved propellant) to be able to perform a rocket-back maneuver to the launch pad.

I estimate that it will be less than 3-tons of added weight penalty on the F9R 2nd stage for the propellant, heat shield, (4) Super Draco thrusters, and landing legs needed to recover and reuse the F9R 2nd stage similar to recovery and reuse of the Dragon-2 spacecraft.

This means that a fully reusable F9R that reuses both its 1st and 2nd stages could carry 10-tons to LEO and less than 1-ton to GTO.

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I've often ran into problems with this general type of thinking for gravity losses. 1,500 m/s is a tempting figure to use, but its clearly flawed and far too high. If you do a Hohmann Transfer calculation you get something closer to 120 m/s. But which model is more accurate? For climb to orbit, its a mess of a combination of them. After first stage, I expect that air and gravity drag will be small, so I would endorse a ballpark calc of 10 or 9.5 - (7.9-2), for about 3.8 km/s contribution. That's what you wound up with anyway. –  AlanSE May 24 at 17:41
    
citation for the mass of the landing legs? –  aramis May 24 at 18:45
    
    
    

You're missing something obvious: Air friction.

The terminal velocity, that is the speed, the rocket would have when air friction and gravity are in balance, is a small fraction of orbital velocities in a few kilometers altitude. So just by reentering the atmosphere, they kill almost all of their momentum.

After reentry, they may also use aerodynamic forces to help get back to the launch pad, but I don't know if that will be a significant factor. In any case, while they need to speed the rocket up quickly during launch, they don't need (or want) a lot of speed to return to base.

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I'll do some calcs when I get home, but if they're going to rely on air friction instead of engine thrust for the bulk of the slow-down, then by the time the first stage re-enters the atmosphere for sufficient drag to slow it down, it'll probably be closer to 200km downrange and at an altitude of only about 50-40km. That seems like it'll be quite low, but if my calculations are correct, it'll still have about 3.65 km/s delta-v left to get back to the pad, so maybe this will do. –  user7388 May 7 at 14:54
    
Of course air friction is a two way street. Yes it will help slow it down, but unpowered reentry has been shown to be destructive to the stage on the first set of flights. So it is a balancing act. Use friction to your advantage. But control it so you can survive it. –  geoffc May 7 at 15:16
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After doing a quick test in the Orbiter sim it looks like using friction is a complete no-go if the F9 first stage has any hope of getting back to the launch pad. Friction substantial enough to produce any sort of slow-down only occurs at around 45km altitude, but even worse, by that time the first stage is more than 1000km from the launch pad! –  user7388 May 7 at 18:29
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But if launched from "Boca Chica Beach" (Brownsville) in Texas and landed in Florida, about 1500 km to the east, then your calculation show an advantage for SpaceX F9r, doesn't it? –  LocalFluff May 8 at 11:00
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Yes, that definitely sounds like a plausible solution, but then they'll have to contend with 1) Strenuous re-entry conditions and 2) Have the FAA (or some similar body) give them the go-ahead to allow the F9 to operate close to inhabited areas (you don't exactly want a rocket to land in your back yard if something goes awry during its ascent stage :P) –  user7388 May 8 at 11:36

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