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Has anyone here ever done enough back-of-the-envelope math to come up with a rough estimate of the maximum landing range of the F9 first stage? I'm talking unladen-swallow-quality math here -- trying to think through just how far F9 is from BFR-like point-to-point ops today, assuming no payload and from a technical standpoint only, ignoring permitting, landing zone availability, etc. (Seems like someone would have discussed this in public already, but I'm not finding it.)

I'm trying to get my head around the math, and I'm realizing that (a) it's been a long time, and (b) it's all about managing re-entry energy, I think...

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    $\begingroup$ if you're happy with the answer below, could you mark it as correct? :) $\endgroup$ – Declan Murphy Oct 9 '17 at 19:20
  • $\begingroup$ Good job with flightclub.io. Forwarding. $\endgroup$ – stevegt Oct 11 '17 at 17:14
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I agree with @OuNelson Mangela on the 4,000km figure.

I got 4,052km on flightclub.io, but it was a bit of a struggle. I had to glide quite a bit between entry and landing burns to bleed off velocity that I didn't have the fuel to remove propulsively, and that extended my range quite a bit.

That said, there may be flight profiles that are more efficient than mine - for example my profile stays quite low in the atmosphere, as I needed to execute the pitch kick quite early to get as much downrange velocity as I could before MECO.

Maybe there is a profile that executes a high-thrust, quite vertical burn to get out of the atmosphere, coasts to near-apogee and then executes a second burn, free from atmospheric and gravity drag to get the downrange distance. Feel free to play around yourself.

To glide between entry and landing, remember that you're travelling engines first, so you want to make sure your pitch angle has a lower magnitude than your velocity angle (i.e you have a positive angle of attack). The larger that angle of attack (up to ~15$^{\circ}$), the more lift you'll experience and the more velocity you'll bleed off before your landing burn.

The flight profile I built can be viewed here and the simulation results are here. I've included some plots of my results below.

  • Flight Profile (note the axes are not to scale)

Flight Profile (note the axis scales)

  • Altitude vs. Time (Booster maintains altitude for longer thanks to the glide)

Altitude vs. Time

  • Angle of Attack vs. Time (note the positive angle of attack to create aerodynamic lift between entry and landing burns)

Angle of Attack vs. Time

  • Booster Phasespace

Booster Phasespace


Updated Answer to keep within Falcon 9 acceleration limit. I've assumed that limit is 6g here.

This was actually a killer. I only made it to ~1,750km downrange. 5km/s of my $\Delta V$ was used in the initial burn which began to heavily throttle about 30s before MECO to keep within 6gs acceleration. This incurred pretty heavy gravity losses so our MECO velocity was only 3.6km/s.

The entry burn needed to be extremely long for us to calm down to a manageable speed entering the atmosphere. Specifically, I used ~2.3km/s of $\Delta V$ for the entry burn but the final 10 seconds of this were also throttled to stay within limits so extra gravity losses were incurred here too. After entry burn cutoff, the atmosphere burned off ~1.3km/s of velocity and then the landing burn had enough propellant remaining to finish the job.

The flight profile I used can be viewed here and the results of the simulation can be viewed here

Here are the corresponding plots to the previous simulation:

  • Flight Profile (note the axes are not to scale)

Flight Profile

  • Altitude vs. Time

Altitude vs. Time

  • Angle of Attack vs. Time

Angle of Attack

  • Booster Phasespace

Booster Phasespace

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  • $\begingroup$ I'm having difficulty understanding the reentry velocity vs altitude. Did you actually just burn up long before landing? A real F9 could not unload much of it's 6,500 m/s velocity using atmospheric friction. You need to use propulsion for that, or invent a heat shield. But kudos and +1 for using a real mathematical tool to address the question! $\endgroup$ – uhoh Oct 5 '17 at 10:18
  • $\begingroup$ @uhoh The glide maintains the booster's altitude over a long period of time thus giving drag more time to do it's work - i.e it uses lift to convert its radial velocity into downrange velocity, then the atmosphere slowly kills that downrange velocity. So that deceleration happened over the course of ~4 minutes and the average deceleration wasn't too bad - however it did spike at 11 Gs which would be pretty uncomfortable :P So yes, I agree it's a lot to ask. Take a look at the results page for more info on altitude, velocity, acceleration and aero pressure over time $\endgroup$ – Declan Murphy Oct 5 '17 at 10:31
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    $\begingroup$ Not to mention that 25 gs on the way up, but don't tell anyone about that. Flight Club lets you specify a maximum deceleration a vehicle can take when you're building it, and it automatically throttles to not go any higher than that - but I disabled that for this demo :) Ha, I didn't realise I had a reputation. Where have you come across my name before?! $\endgroup$ – Declan Murphy Oct 5 '17 at 10:38
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    $\begingroup$ I'm a big flightclub.io fan! $\endgroup$ – uhoh Oct 5 '17 at 10:41
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    $\begingroup$ @uhoh Updated. I didn't think there would be so much of an ill effect on performance, but the proof is in the pudding. $\endgroup$ – Declan Murphy Oct 7 '17 at 14:59
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I can't remenber the entire math/base values and am rounding the values abit. We have a rocket with about 300(SL) to 340sec(vac) and a dry mass of 5 or 6% Can't remenber for sure. lets assume 6% that gives an "avg" of 327sec in a relativelly shallow launch profile.

ln 16.6*327=9km/s of delta v(calculator says 9.2,LOL)

A single stage migth be able to reach orbit or near orbit, though without any payload. Not very good for sensing cargo to LEO but plenty reach anywhere on earth even shooting for a retrograde suborbit.

The minimun to reach anywhere on Earth is no more tham 8km/s if I'm not mistaken.

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    $\begingroup$ I'm not sure this is a reasonable calculation. The goal is to safely return to Earth. You need to save some of that fuel to slow down before re-entry! A F9 1st stage can not re-enter the atmosphere at orbital velocity. Also there may be a (very roughly) 1 km/s loss from gravity + drag. $\endgroup$ – uhoh Oct 4 '17 at 8:38
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    $\begingroup$ @uhoh - I was counting for the gravity and aerodinamic losses though I din't count for the whole landing angle. With 9,2 km/syou should be able to acelerate to about 5 km per second and range of about 2 or 3 thousand kilometers. That is counting curent velocities that i calculate the booster can shed without overheating. Though now the calculations have significant more error. it migth be as much as 4000km. $\endgroup$ – OuNelson Mangela Oct 4 '17 at 10:00
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    $\begingroup$ @OuNelsonMangela I agree on the 4,000km. I got 4,052km on flightclub.io, but it was a bit of a struggle. I had to glide quite a bit between entry and landing burns to bleed off velocity that I didn't have fuel for, and that extended my range quite a bit. My results are here (the flight profile URL is too many characters to post in a comment) $\endgroup$ – Declan Murphy Oct 4 '17 at 15:56
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    $\begingroup$ Declan Murphy, if you simulated this with a good resolution then I'd consider your comment an Answer. Please consider posting it as such. Your simulated flight has to be at least as good an answer as any back-of-the-envelope math solution. $\endgroup$ – Kengineer Oct 4 '17 at 19:44

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