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Planet Earthtoo saw that Earth could put a person in orbit, so they wanted to go to space too.

The planet Earthtoo is twice the diameter of Earth, with the same internal structure - the average density is the same as Earth so the total mass is 8 times that of Earth, and thus the surface gravity is twice that of earth.

LEtooO (Low Earthtoo Orbit) is lower than that on earth because even though the surface pressure is twice that of Earth, the scale height is half. However, Tooian scientists have decided that 200km is their target height for the first tootronaut to safely orbit.

Earthtoo also has a 24 hour day. They have no mountains, but a nice coastline on the equator from which to launch and take advantage of the rotation speed.

Tooian rocket scientists have perfected a conventional rocket engine with an Isp of 500 seconds independent of ambient pressure. They can be scaled in size.

Because of their deep admiration of Earth astronaut Don Pettit, they have studied his discussion of The Tyranny of gravity and worked and worked until they could build each stage of their rocket with 96% of its mass as propellant.

The mass of the orbiter with the tootronaut, life support, and substantial re-entry protection (twice the orbital velocity, twice the gravitational force) was an amazingly low 1000kg. They've been watching earth very carefully and took the best technologies. They sometimes call it the "Flying Tile" because of the substantial use of ceramic tile material.

Question: Based on the parameters below and situation described above, what is a rocket design that can put the tootronaut and orbiter into LEtooO? Please choose a widely accepted and recognized spaceflight simulator (Kerbal could be OK for example). Show enough of the answer that other people can independently run your design into orbit and verify that "yep, it works." How big is this rocket?


Use the following values:

radius of planet:   12740 km
rotation period:    24 hours
launch location:    equator
surface gravity:    19.6 m/s^2
atmosphere density 
    at surface:     2.4 kg/m^3
scale height:       3.8 km

mass of payload:    1000 kg
specific impulse*:  500 sec (any size engine, any ambient pressure)
stage dry mass:     0.04 of total mass when stage is fully fueled.

altitude of orbit:  200 km
orbital velocity:   15.8 km/s or thereabouts, for a circular orbit
    (derived)
orbit inclination:  0 degrees

(*) remember to use 9.8 and NOT 19.6 if you convert Isp to effective thrust velocity!
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  • $\begingroup$ My other question is still unanswered (with sufficient proof), and there is some question whether astronaut Don Pettit's Tyranny essay's value of 1.5 earth applies, or if Tsiolkovsky's rocket equation is enough to say that no matter how heavy, there's some size rocket that will get you to orbit. So to break the logjam, I'm drawing a line in the sand. Can a conventional rocket (slightly spiced up) of some size get a 1000kg spacecraft into orbit around a twice-the-size-but-same-average-density planet? $\endgroup$ – uhoh Jul 24 '16 at 8:33
  • $\begingroup$ This is very similar to space.stackexchange.com/questions/17423/…, is there a reason we shouldn't close this as a duplicate? $\endgroup$ – Hobbes Jul 24 '16 at 9:42
  • $\begingroup$ @Hobbes Yes, because it is not a duplicate for one. This question asks for the actual size of the rocket that would be necessary. It asks for the specifics of the staging using some simulator that others can try and verify. I'm trying to get a quantified, verifiable, numerical answer. What you are proposing will create yet another pointer to a location that doesn't have what I'm asking for. A verifiable rocket design and simulation to orbit will put this to rest. So at least hold off for a bit please! $\endgroup$ – uhoh Jul 24 '16 at 9:46
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    $\begingroup$ Incidentally, Musk saying "KSP is awesome" doesn't indicate that SpaceX uses it as a design tool. It is certainly useful for developing an understanding of principles, but that's about it. $\endgroup$ – Russell Borogove Jul 24 '16 at 15:36
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    $\begingroup$ shrug KSP is heavily mod'able. There are a few 'autopilot' mods. There are also realism mods that give both more realistic gravity and atmosphere. It would be totally unsurprising if it was used as the test framework for prototyping the vertical landing system - developing a physics simulator of similar complexity would cost a fortune, meanwhile writing a mod that creates a rocket of Falcon 9 capabilities is a couple afternoons of work. Then a mod which connects the game input/output data with the control software... $\endgroup$ – SF. Jul 25 '16 at 2:18
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Pettit's article doesn't mention the advantage of staging at all. All he's actually saying is that a hydrogen fueled single-stage-to-orbit launcher can't work on a planet 50% larger than Earth. Staging doesn't invalidate the Tsiolkovsky equation but it does work around the need to attain extreme propellant mass fractions in a single stage.

As Hohmannfan notes, the Isp and mass ratio assumptions in the question as written are pretty unrealistic, but I think the Tooians can do this with Apollo-era technology and a 4-stage rocket 4 times the size of a Saturn V.

I think the gravity losses are a bigger concern, so I aimed for 20km/s of total ∆v.

Stages:

  • Payload: Stripped Mercury, 1 t
  • Stage 4: Basically a shortened Centaur, 2 RL10-A3, 13t, 6038 m/s
  • Stage 3: 3x J-2, 135t, 5535 m/s
  • Stage 2: 5x M-1, 1500t, 5462 m/s
  • Stage 1: 40x F-1, 9500t, 3261 m/s

Total 11149 tons -- Saturn V is about 3000.

The fuel fractions on the stages are 80%-84%, since I assume the larger structure and higher gravity are concerns. Generally it needs twice as much engine per mass as Terran rockets, which also contributes to dry mass. Here's my worksheet:

enter image description here

The three upper stages are smaller than the three stages of the Saturn V launcher; there's no question that these could be built. The M-1 engine was never completed and flown but got well into development, and there's no reason to think it wouldn't have been successful. It could be replaced on a 6:1 basis with J-2s if not. The first stage is certainly hefty. Simply finding enough stage-base area to mount so many engines requires that the stages be short and squat (or perhaps conical, like the Russian N-1 design); this means that drag will be of greater relative concern than on Earth.

The max accel values assume that each stage continues burning at full thrust until fuel is expended. Saturn V shut down center engines partway through the burns on the first two stages in order to reduce g-force on the crew, peaking at about 4g. I assume Tooians are a little tougher, but if necessary, this rocket can do something similar. Obviously the first stage needs better than 2g initial acceleration (=1:1 TWR) for liftoff. The initial acceleration of the other stages may seem low, but note that Saturn V second stage lights up at only 0.8g and third stage about 0.55g, while the Tooian rocket maintains better than 1:1 TWR at all points until the last stage.

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    $\begingroup$ I thought it could be fun to try to stack together a launcher using existing stages, but I did not do it. I love the way you solved it! $\endgroup$ – Hohmannfan Jul 24 '16 at 16:37
  • $\begingroup$ Don't forget to modify the Mercury capsule to handle double the re-entry velocity and far more rapidly increasing air density. It's not part of this question, but they don't call it the flying tile for nothing. However, it is now part of this followup question. I really like your approach too! $\endgroup$ – uhoh Jul 25 '16 at 2:52
  • $\begingroup$ Also - I don't see the word "single" or the acronym "SSTO" anywhere in the Tyranny essay, but I do see the sentence "If our planet was 50% larger in diameter, we would not be able to venture into space, at least using rockets for transport." $\endgroup$ – uhoh Jul 25 '16 at 3:02
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    $\begingroup$ But that conclusion is obviously wrong if you take staging into account. $\endgroup$ – Russell Borogove Jul 25 '16 at 3:03
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    $\begingroup$ As for the capsule, "once the rockets go up, who cares how they come down? / that's not my department, says Wehrner Von Braun". $\endgroup$ – Russell Borogove Jul 25 '16 at 3:08
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First, we need to find what additional $\Delta v$ the rocket needs to achieve orbit. Earth LEO velocity is for instance 7.8 km/s, but the real $\Delta v$ needed is somewhere around 9-9.5km/s because of drag and gravity losses. That is an additional 1.5km/s. The first thing I notice is that Earthtoo has a denser atmosphere, but that the scale height is lower. Here is a plot comparing the atmosphere to ours:

atmosphere of Earthtoo

After just a few kilometres, the density is in fact lower. Drag is more important when the velocity starts to get high, so I will say that an Earthtoo rocket looses less $\Delta v$ due to it. That should compensate a little for the higher gravity losses. I want more thrust too, so I chose larger engines to compensate for the rest, increasing my dry mass share to 0.05. For a worst case scenario, I then design my rocket for a $\Delta v$ of 18 km/s.

Here is my design:

I have 3 stages, each of them 3 times more massive than all the mass they carry.

Payload:    1000kg
3rd stage:  4000kg ( 200kg dry mass)
2nd stage: 15000kg ( 750kg dry mass)
1st stage: 60000kg (3000kg dry mass)
Total:     80000kg

The idea is that the mass ratio for each expended stage is then $\frac{1+3}{1+3 \cdot 0.05} = 3.48$.

The $\Delta v$ added for each of the three stages is then:

$$\Delta v = \ln{3.48} \cdot 500s \cdot 9.81m/s^2 = 6120 m/s$$

That is 18360m/s in total, slightly more than needed.

I can not see the need to run this in a sophisticated spaceflight simulator, as some of the given parameters are highly unlikely. For instance, a performance of 500s independent of ambient pressure is extremely optimistic, as most launchers $I_{sp}$ at sea level is around 300s, and an atmosphere twice as dense does not help at all. Secondly, complete rocket stages seldom have a better total-to-dry mass ratio than about 15. (0.067), so even the more conservative figure I used (20 instead of 25) is fiction.

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  • $\begingroup$ Great!! It's a hypothetical situation, and I wanted to make initial conditions as simple as possible. Fixed Isp of 500 sec is better than what we have on earth, but not so much better that it has to be a nuclear rocket or something. Maybe they use liquid aluminum or something else never considered on earth, and an articulated nozzles and other things to adjust for ambient pressure. It's meant to be a simple starting point. $\endgroup$ – uhoh Jul 24 '16 at 16:27
  • $\begingroup$ If I gave engine parameters that are typical technologies now, I'd get slammed for saying it was unrealistic because they would have worked harder. $\endgroup$ – uhoh Jul 24 '16 at 16:28
  • $\begingroup$ I looked at the Elon Musk AMA Reddit and I think he was serious when he said that he thinks that their upper stages at least could some day reach 0.96 or 0.97 mass fraction. I just figured, like Herz Rent-a-Car's ancient slogan, Earthtoo is #2 so they "...will try harder." But wow, seems you didin't even need it! $\endgroup$ – uhoh Jul 24 '16 at 17:13
  • $\begingroup$ Would you be interested in running a simulator if you used realistic values instead? I chose the "unrealistic" ones because I thought this was going to be very difficult, especially with the Petit 1.5xEarth prediction out there. If not, just a rough estimate of altitude, velocity and time of the two stage separations and orbital insertion? I'm curious if the whole thing is actually faster or slower than LEO on Earth! $\endgroup$ – uhoh Jul 28 '16 at 11:49

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