How much bigger could Earth be before rockets wouldn’t work?

asked Mar 9, 2016 at 7:45

uhohuhoh

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Since direct boosts in delta-v need rapid boosts in mass, little modifications to the presumptions you make about fuel tank structural mass and engine thrust-to-weight ratio start to make huge modifications in the last size of the rocket.

If you’re getting off a 3.6 g world with a 7-stage rocket, the distinction in between 88% fuel portion and 92% fuel portion yields about a 10:1 distinction in the overall mass of the rocket.

I do not believe it’s truly sensible to talk about supreme theoretical limitations; too numerous engineering aspects are included.

Locking down a lot of variables, I can inform you what kind of rocket you ‘d require for an offered surface area g. Let’s make these presumptions:

• We are positioning 1 lots of payload into low planetary orbit.
• Needed delta-v to reach orbit, consisting of climatic and gravity losses, is 10,000 m/s per surface area g. Seems to hold for Earth, Mars, and the “Earthtoo” which was gone over in another Q/A
• We can construct rocket phases of approximate size, with a tankage propellant portion of 90%; the rocket phase mass is the tank mass plus the engine mass– ullage rockets, interstage, and so on is all handwaved out.
• We have a boundless supply of Apollo-era rocket engines: RL-10, J-2, M-1, H-1, and F-1.
• First-stage TWR at ignition need to be at least 1.2 (relative to regional gravity)
• Middle-stage TWR at ignition need to be at least 0.8
• Final-stage TWR at ignition need to be at least 0.5

Provided those presumptions, here is a table of surface area gravity, phase count, first-stage engines, and overall rocket mass.

Surface                         First        Total       Saturn V 
Gravity   Stages                Stage      Mass, t     Equivalent
 0.5           2             1x RL-10          4.5
 1.0           3             1x   H-1         49.4          0.02
 1.5           3             1x   F-1        249.2           0.1
 2.0           4             5x   F-1       1329.0           0.5
 2.5           5            40x   F-1       8500.9             3
 3.0           6           274x   F-1      50722.2            17
 3.5           7          2069x   F-1     331430.9           100
 4.0           8         20422x   F-1    2836598.4           950
 4.5           8        392098x   F-1   47 million         15000
 5.0           9    3.5 million   F-1  391 million        130000
 6.0          11    400 million   F-1   38 billion      millions
10.0          18        2.88e19   F-1      1.65e21  quadrillions

Up above 10g, something actually intriguing takes place that is sort of a theoretical limitation. The mass of the rocket reaches a quantifiable portion of the mass of the whole world it’s releasing from.

At 10.3 g, rocket mass is 0.035 of the mass of the world. 10.4 g, rocket mass is one fifth of the mass of the world. This does not in fact change the ∆ v requirement– we’re going into orbit around the rocket/planet barycenter! At 10.47 g, the rocket is the world, and we’re … simply … chewing it up totally, crushing it in a dust cloud broadening at 4km/s.

These severe conclusions seem supported by this individually obtained paperwhich checks out some other associated elements of super-Earth-based chemical rockets.

Another factor to consider just recently raised by user @uhoh is that as the direct scale of a provided rocket phase boosts, its mass, and therefore the needed thrust force to raise it, increases by the cube of the scale, however the location readily available at the base of the rocket to install engines increases just by the square of the scale; this issue is made worse here by the increasing surface area gravity. The Saturn V was practically at the point where this relation begins to end up being troublesome; the outboard engines on its very first phase are installed at the very edge of the phase in order to include their nozzles to gimbal.

Strong rockets do not have the very same dimensional restrictions, and have great thrust-to-weight and thrust-to-cost ratios, so they’re most likely most likely to be utilized in lower phases for these large rockets.

Phases much bigger than the Saturn V very first phase would require to resolve this with some mix of being much shorter and squatter, or jeopardizing engine gimbal variety, or installing engines in pods surrounding the tankage, and there may be relatively difficult engineering limitations eventually for those factors. At the 3g mark, for instance, the 274 first-stage engines would need a phase about 90 meters in size and 9 meters high, at which point the engineering inadequacies related to the fuel tank percentages will be ending up being major.

Let us look at the rocket formula:

$$ Delta v= ln left( frac m_0 best) v_e$$

That informs just how much a rocket can alter its speed (the $ Delta v$. The requirements for reaching a greater speed for a very little orbit would increase on your much heavier Earth. (For consistent density it is proportional to the radius.)

How can we increase the $ Delta v$ of the rocket to maintain? We can increase the exhaust speed, $v_e$of the engine, however that cut-off is around 5000 m/s for chemical engines. The other thing we might do is increasing the mass ratio of the rocket $ left( frac m_f right)$That is troublesome too, as we can not actually make the fuel tanks out of soap bubbles. Staging is the alternative left, you might position a huge rocket under a little rocket to get a bit more modification in speed. You are getting a direct advantage for a rapid cost.

As an example, the Saturn V rocket entered into LEO (~ 9000 m/s), sent out a payload towards the Moon (3120 m/s), the service module slowed the stack into LMO (820 m/s), and lastly the LM landed and removed once again (2 * 1720 m/s). There are still some unused fuel left in the service module then, so let us simply call the overall $ Delta v$ of the Saturn V/Apollo 17 km/s. That is less than the requirements for a 2x radius Earth. The Apollo program was quite pricey [citation needed]so it might take a while before a country of a 2x Earth world tries to go into orbit. The limitation is, as you specify, the extremely low payload ratio.

Another factor to consider is the increased surface area gravity. (That scales linearly with size at continuous density). That needs the rocket to have a greater thrust to weight ratio, which will increase the dry mass, minimizing the possible $ Delta v$(It likewise increases gravity losses, however that is primarily compensated by the lower scale height of the world, minimizing drag losses).

Ultimately, the gravity is so high that even the most effective engine can not raise itself from the ground. That a minimum of is a conclusive limitation.

A more theoretical factor to consider, is $ Delta v$ requirements really a limited limitation?

Remarkably, it is not. Remember what I stated about staging previously: “you are getting a direct advantage for a rapid expenditure”. There is not restrict to what we can use up! Think about the following circumstance: We include increasingly more phases at the bottom of the rocket, each of them has the exact same mass as all the phases on top of it. Burning each of them offers the very same mass ratio in between previously and after, for that reason each of them are providing the very same quantity of $ Delta v$To include 10 times that quantity, you require 10 phases each doubling the mass. To include 100 times that quantity, you require to double a hundred times. The mass grows unbelievably quickly, even doubling 10 times are over a thousand times more. Why ought to we stop:-RRB-

Can we actually continue to include significantly bigger phases for ever?

After a while, other issues appear. : Rockets are long and thin, to decrease drag. That shape can not be kept for large rockets. The factor not is the square cube lawSaving the very same dimensional percentages, a rocket two times the height has 8 times more mass. The base location of the rocket has actually just increased 4 times. That implies that each system of location needs to support more mass. Eventually, even the greatest products need to quit, and you should quit the standard rocket shape in favour of a broader base. That includes a lot to the drag! Issues like that are going to continue to appear:

“More mass implies more issues, greatly more mass ways greatly more issues.”

Summed up:

A contemporary style, bigger rocket than the Saturn V, with adjustments to increase the T/W ratio might most likely make it to orbit on a 2x radius, 8x mass Earth. That is an expediency limitation, rockets that are extremely much bigger might have a couple of km/s additional $ Delta v$however that does not change the numbers a lot. In theory however, rockets can grow up until the drag stops them, or the engines can no longer raise even themselves.

Or maybe you eventually wish to utilize the readily available resources of the world to introduce a single rocket to orbit.

note: I’ve accepted a response 2.5 years back. This paper was released just recently so I believed I would include this extra response because it might be a fascinating recommendation for future readers.

The Space.com short article No chance Out? Aliens on ‘Super-Earth’ Planets May Be Trapped by Gravity links to Michael Hippke’s ArXiv preprint Spaceflight from Super-Earths is challenging

While the estimation is based upon escape speed instead of LSEO (Low Super-Earth Orbit) the conclusion is comparable, the issue is rapid and it gets actually challenging rapidly.

The author utilizes the example of the world Keppler-20b (see likewise hereand although there is some unpredictability, the world’s size is approximately 1.9 that of earth, and it’s mass is nearly 10 times that of Earth.

For a mass ratio of 83, the minimum rocket (1 t to $v _ $would bring 9,000 t of fuel on Kepler-20b, which is 3 × bigger than a Saturn V (which raised 45 t). To raise a better payload of 6.2 t as needed for the James Webb Space Telescope on Kepler-20 b, the fuel mass would increase to 55,000 t, about the mass of the biggest ocean battleships. For a classical Apollo moon objective (45 t), the rocket would require to be significantly bigger, ∼ 400,000 t. This is of order the mass of the Pyramid of Cheops, and is most likely a reasonable limitation for chemical rockets relating to expense restrictions. (focus included)

Not a planetological exposition in sight so, I’ll include my 2 cents to this rather theoretical conversation.

Among exoplanetologists, the agreement has actually emerged that 1.6 Earth radii and 5 Earth masses is most likely to be the ceiling to rocky worldsSimulations have actually revealed that above these figures, the bodies establish progressively Mini-Neptune like attributes. This suggests extremely thick Helium Hydrogen environments and squashing surface area pressure.

Because Michael Hippke’s somewhat whimsical paper was referenced in among the responses it appears proper to discuss Ocean worlds at Super Earth masses. Ocean worlds provide a host of habitability obstacles consisting of a scarceness of particular life vital components like phosphorus, absence of volcanism, no water rock user interface due to high pressure ice on the marine flooring and others. These conditions will likely restrict or perhaps avoid the facility of the dynamic prebiotic chemical environments that are needed for biogenesis.

If the very first presumption is true, the greatest gravity on a possibly habitable world will not go beyond around 2.5 g.(edit: and therefore making it not rather so hard to reach orbit with chemical rockets as would have held true with a greater g worth)

Excellent responses have actually been provided, however among the significant styles is they presume a repaired damp to dry mass ratio of 10:1 (ish). The reason is:

• You require to repair this as: there are no significant responses with without a worth and, which worth undergoes engineering subtleties, which are tough to manage.

• 10:1 is an excellent choice. (We can’t do far better than this and still have whatever work so it appears practical to stick at this)

The issue is that’s the limitation of what we can make work in the worldA great deal of the dry mass of a rocket is either:

• straight connected to the thrust-to-mass ratio (i.e. number/size of engines)

• indirectly associated to TMR (i.e. supports the structural loads)

Keep in mind, to keep gravity loses comparable in practice the velocities required, thus TMR, is direct with the surface area gravity. So is a part of the wet/dry mass ratio.

When we take that into factor to consider things look a lot bleaker for the high g super-earths getting something into orbit utilizing chemical rockets.

The real numbers here are a little tough to understand, however if 5g world causes a rocket with a w/d mass ratio of 5 to 1 (which I believe has to do with ideal however …), you’re gazing down the barrel of a $10 ^ 20 $t type figure for launch mass. To put that into point of view, the ‘moon rocket’ is no longer a great contrast. That’s the mass of the moon it got to.

Theoretical limitation? I ‘d state so

At that mass things begin deviating for the ‘XKCD’. Forget the useful concerns they’re plainly long addressed “moon-sized-anything”. We struck cold difficult theoretical limitations.
You begin needing to handle your own gravity

Those useful problems are huge ones even if we laugh a little ‘engineering’ issues (like cash, and where we may discover $10 ^ $t of aerospace grade products). That’s the sort of size that when you’re made out something strong and are currently drifting in area under 0G, you warping under you own gravity into a ball. Attempting to make that out of primarily liquid fuel and subject it to 5-10g …, you’re not remaining the shape you began. Does not matter what mass-ratio ‘struck’ you want to take. We’ve got this far, we aren’t going to let an absence of unobtainium stop us.

No the genuine hard limitation is how being so heavy results your exhaust speed. At the threat of getting too meta here if you’re heavy enough, its challenging to get things to come apart from you. It uses to world sized rockets as much as it does worlds.

If you’re a couple of million kilos, your ‘exhaust speed’ is the speed you can get your propellant to get to. If you have more mass than the moon, your propellant will have lost a great deal of momentum by the time it’s left your gravitational impact. And this is the fate our rocket fulfills. LOX/H2 has an exhaust speed of about $4,400 ms ^ -1 $about as excellent as we can do. Let’s simply state our moon-sized rocket has the density of the moon too, therefore has a comparable escape speed of $2,380 ms ^ $The beneficial exhaust speed of our rocket (preliminary less escape) is less than half. Half the delta-v. You will not be going to area to day.

“Ok”, I hear you state, “that simply indicate’s you can’t go to area in that rocket.”, “How about a larger one?”. Well “No”. This is another among those “Even if whatever sort of worked as previously, you desire go two times as quick, which is going to be a lot more mass.” type issues. Other than now we actually can’t simply take the “make in 10 orders of magnitude larger” technique. Apart from the truth that our rocket is now lot larger than the world which implies we could not potentially build it, now we have no possibility of utilizing chemical rockets to move us anywhere. To acquire any momentum we require to chuck something out of our gravity well, and the exhaust speed of chemical rockets do not make when we are this huge. We are now really stuck.

Wait: straight the exhaust does not make it out, however I question if you might attempt various method of getting mass out of an extremely deep gravity well. Should not be too tough. Even if it was just a bit, we might constantly simply scale it up …

On a useful engineering side of things. Eventually you are restricted by exhaust speed. In theory you can constantly simply make a larger engine, larger tanks, and so on. Extremely pricey, however possible. This would appear to set the genuine limitation to material strength. Product strength is most likely to provide before the gravity wells pull goes beyond the exhaust speed of even reasonably modern-day fuels.

LF+LOX usually has an exhaust speed of around 4,400 m/s. Which will battle as much as 448 G of gravity. Actually more than the sun. Almost nevertheless much less than that. Size of the world itself provides no genuine offer killers, it simply makes the payloads mass portion really VERY low.

At some time though other innovations, like nuke drives (https://en.wikipedia.org/wiki/Project_Orion_(nuclear_propulsion)end up being the only possible budget friendly method off the world.