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trace: Apparently the Tsiolkovsky rocket equation doesn’t truly say that you may launch a standard rocket into orbit round an arbitrarily giant and big physique.
I am in search of a quantity based mostly on scaling the earth radius and sustaining the identical common density. Should attain LEO, which additionally will get sooner because the planet grows. Don Pettit’s Tyranny talked about in this nice answer is enjoyable, however doesn’t current sufficient math.
On this Earth, rockets barely work. Payloads can solely be a couple of % of the full mass for LEO, and fewer than one % for deep house.
If we outline barely heavier Earths, say Earth_{1.1}, Earth_{1.2}… the place the radii had been 1.1, 1.2, and so forth. instances that of Earth and the lots had been 1.1^{3}, 1.2^{3}, and so forth. instances the Earth’s mass (in different phrases similar common density, similar “iron/rock ratio”) what occurs? Is there some level the place chemical rockets merely will now not be capable to put issues in house, or does the payload mass merely turn into ridiculously tiny? If there’s a cutoff, is it totally different for LEO and deep house?
For our functions, let’s not discover different or hybrid launch methods or enhance methods (reminiscent of balloons, planes, laser beams, house elevators and so forth.). Simply persist with chemical propellant rockets.
edit: here’s a information. So for a scaling issue $f$:
$$
r = f r_{earth}
$$
$$
m = f^3 m_{earth}
$$
$$
g = G frac{m}{r^2} = frac{f^3}{f^2}g_{earth} = f g_{earth}
$$
$$
H = frac{kT}{gm_{molecule}} = f^{1}H_{earth}
$$
We catch a bit break right here. Assuming similar floor ambiance composition, temperature and strain (STP), the scale height H truly decreases with growing $f$. (If we had been “world builders” we must always most likely improve strain to get extra oxygen wanted for transferring within the increased gravity, however that is a different Stack Exchange.)
So far as LEO altitude is anxious (thanks @Lex for catching that) one would possibly outline it as the identical variety of scale heights as could be on Earth. That is probably not so helpful as a result of the density profiles of the bits of the ambiance answerable for drag (Thermosphere and Exosphere are affected by many phenomenon, together with the photo voltaic wind, and do not scale in any respect just like the decrease layers. Nonetheless for historic causes I will depart the next, as it’s not important to the query:
$$
h_{LEO} = h_{LEOearth} frac{H}{H_{earth}} = f^{1} h_{LEOearth}
$$
$$
v_{LEO}=fv_{LEOearth}
$$
The LEO interval is unbiased of the dimensions of a planet, if the common density is mounted. Nevertheless, the speed of LEO does scale with radius!
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As a result of linear will increase in deltav require exponential will increase in mass, small adjustments to the assumptions you make about gasoline tank structural mass and engine thrusttoweight ratio begin to make very giant adjustments within the remaining measurement of the rocket.
For instance, when you’re getting off a 3.6g planet with a 7stage rocket, the distinction between 88% gasoline fraction and 92% gasoline fraction yields a couple of 10:1 distinction within the whole mass of the rocket.
So I do not assume it is actually cheap to speak about final theoretical limits; too many engineering components are concerned.
Locking down quite a lot of variables, I can inform you what sort of rocket you’d want for a given floor g, although. Let’s make these assumptions:
 We’re putting 1 ton of payload into low planetary orbit.
 Required deltav to succeed in orbit, together with atmospheric and gravity losses, is 10,000m/s per floor g. Appears to carry for Earth, Mars, and the “Earthtoo” which was mentioned in another Q/A.
 We will construct rocket phases of arbitrary measurement, with a tankage propellant fraction of 90%; the rocket stage mass is the tank mass plus the engine mass — ullage rockets, interstage, and so forth. is all handwaved out.
 We now have an infinite provide of Apolloera rocket engines: RL10, J2, M1, H1, and F1.
 Firststage TWR at ignition should be a minimum of 1.2 (relative to native gravity)
 Centerstage TWR at ignition should be a minimum of 0.8
 Ultimatestage TWR at ignition should be a minimum of 0.5
Given these assumptions, here’s a desk of floor gravity, stage depend, firststage engines, and whole rocket mass.
Floor First Complete Saturn V
Gravity Levels Stage Mass, t Equal
0.5 2 1x RL10 4.5
1.0 3 1x H1 49.4 0.02
1.5 3 1x F1 249.2 0.1
2.0 4 5x F1 1329.0 0.5
2.5 5 40x F1 8500.9 3
3.0 6 274x F1 50722.2 17
3.5 7 2069x F1 331430.9 100
4.0 8 20422x F1 2836598.4 950
4.5 8 392098x F1 47 million 15000
5.0 9 3.5 million F1 391 million 130000
6.0 11 400 million F1 38 billion hundreds of thousands
10.0 18 2.88e19 F1 1.65e21 quadrillions
Up above 10g, one thing actually attentiongrabbing occurs that’s type of a theoretical restrict. The mass of the rocket reaches a measurable fraction of the mass of all the planet it is launching from.
At 10.3g, rocket mass is 0.035 of the mass of the planet.
10.4g, rocket mass is one fifth of the mass of the planet. This does not truly alter the ∆v requirement — we’re going into orbit across the rocket/planet barycenter!
At 10.47g, the rocket is the planet, and we’re… simply… chewing it up completely, pulverizing it in a mud cloud increasing at 4km/s.
These excessive conclusions look like corroborated by this independently derived paper, which explores another associated facets of superEarthbased chemical rockets.
One other consideration just lately introduced up by person @uhoh is that because the linear scale of a given rocket stage will increase, its mass, and thus the required thrust drive to elevate it, goes up by the dice of the size, however the space out there on the base of the rocket to mount engines goes up solely by the sq. of the size; this downside is made even worse right here by the growing floor gravity. The Saturn V was nearly on the level the place this relation begins to turn into problematic; the outboard engines on its first stage are mounted on the very fringe of the stage with the intention to make room for his or her nozzles to gimbal.
Stable rockets haven’t got the identical dimensional constraints, and have superb thrusttoweight and thrusttocost ratios, in order that they’re most likely extra possible for use in decrease phases for these very giant rockets.
Levels a lot bigger than the Saturn V first stage would wish to handle this with some mixture of being shorter and squatter, or compromising engine gimbal vary, or mounting engines in pods surrounding the tankage, and there is likely to be pretty laborious engineering limits sooner or later for these causes. On the 3g mark, for instance, the 274 firststage engines would require a stage about 90 meters in diameter and 9 meters tall, at which level the engineering inefficiencies related to the gasoline tank proportions will probably be turning into severe.
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answered Jul 30, 2016 at 20:53
Russell BorogoveRussell Borogove
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First, allow us to have a look at the rocket equation:
$$Delta v=ln left(frac{m_0}{m_f}proper)v_e$$
That tells how a lot a rocket can change its velocity (the $Delta v$). The necessities for reaching a better velocity for a minimal orbit would improve in your heavier Earth. (For fixed density it’s proportional to the radius.)
How can we improve the $Delta v$ of the rocket to maintain up? We will improve the exhaust velocity, $v_e$, of the engine, however that cutoff is round 5000 m/s for chemical engines. The opposite factor we may do is growing the mass ratio of the rocket $left(frac{m_0}{m_f}proper)$. That’s problematic too, as we can’t actually make the gasoline tanks out of cleaning soap bubbles. Staging is the choice left, you may place a giant rocket beneath a small rocket to get a bit extra change in velocity. Then you might be getting a linear profit for an exponential expense.
For instance, the Saturn V rocket acquired into LEO (~9000 m/s), despatched a payload in the direction of the Moon (3120 m/s), the service module slowed the stack into LMO (820 m/s), and eventually the LM landed and took off once more (2*1720 m/s). There are nonetheless some unused gasoline left within the service module then, so allow us to simply name the full $Delta v$ of the Saturn V/Apollo 17 km/s. That’s lower than the necessities for a 2x radius Earth. The Apollo program was fairly costly [citation needed], so it might take some time earlier than a nation of a 2x Earth world makes an attempt to enter orbit. The restrict is, as you state, the ridiculously low payload ratio.
One other consideration is the elevated floor gravity. (That scales linearly with diameter at fixed density). That requires the rocket to have a better thrust to weight ratio, and that may improve the dry mass, decreasing the potential $Delta v$. (It additionally will increase gravity losses, however that’s principally compensated by the decrease scale top of the planet, decreasing drag losses).
Finally, the gravity is so excessive that even essentially the most highly effective engine can’t elevate itself from the bottom. That a minimum of is a definitive restrict.
A extra theoretical consideration, is $Delta v$ necessities truly a finite restrict?
Surprisingly, it’s not. Keep in mind what I mentioned about staging earlier: “you might be getting a linear profit for an exponential expense”. However there’s not restrict to what we will expend! Think about the next state of affairs: We add an increasing number of phases on the backside of the rocket, every of them has the identical mass as all of the phases on prime of it. Then burning every of them provides the identical mass ratio between earlier than and after, subsequently every of them are supplying
the identical quantity of $Delta v$. So as to add 10 instances that quantity, you want 10 phases every doubling the mass. So as to add 100 instances that quantity, it’s good to double 100 instances. The mass grows ridiculously quick, even doubling 10 instances are over a thousand instances extra. However why ought to we cease 🙂
However can we actually proceed so as to add exponentially bigger phases for ever?
After some time, different issues present up. For example: Rockets are lengthy and skinny, to reduce drag. That form can’t be stored for very giant rockets. The explanation not is the square cube law. Conserving the identical dimensional proportions, a rocket twice the peak has 8 instances extra mass. However the base space of the rocket has solely elevated 4 instances. That implies that every unit of space has to assist extra mass. Eventually, even the strongest supplies should quit, and you need to quit the normal rocket form in favour of a wider base. That provides lots to the drag! Issues like which can be going to proceed to point out up:
“Extra mass means extra issues, exponentially extra mass means exponentially extra issues.”
Summarized:
A contemporary design, bigger rocket than the Saturn V, with modifications to extend the T/W ratio may most likely make it to orbit on a 2x radius, 8x mass Earth. That could be a feasibility restrict, rockets which can be ridiculously a lot bigger could have a couple of km/s additional $Delta v$, however that doesn’t alter the numbers lots. In idea although, rockets can develop till the drag stops them, or the engines can now not elevate even themselves.
Or maybe you sooner or later need to use the out there assets of the planet to launch a single rocket to orbit.
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word: I’ve accepted a solution 2.5 years in the past. This paper was printed just lately so I believed I might add this supplemental reply since it might be an attentiongrabbing reference for future readers.
The House.com article No Way Out? Aliens on ‘SuperEarth’ Planets May Be Trapped by Gravity hyperlinks to Michael Hippke’s ArXiv preprint Spaceflight from SuperEarths is difficult.
Whereas the calculation is predicated on escape velocity moderately than LSEO (Low TremendousEarth Orbit) the conclusion is comparable, the issue is exponential and it will get actually tough shortly.
The writer makes use of the instance of the planet Keppler20b (see additionally here), and though there’s some uncertainty, the planet’s measurement is roughly 1.9 that of earth, and it is mass is nearly 10 instances that of Earth.
For a mass ratio of 83, the minimal rocket (1 t to $v_{esc}$) would carry 9,000 t of gasoline on Kepler20b, which is 3× bigger than a Saturn V (which lifted 45 t). To
elevate a extra helpful payload of 6.2 t as required for the James Webb House Telescope on Kepler20 b, the gasoline mass would improve to 55,000 t, in regards to the mass of the most important ocean battleships. For a classical Apollo moon mission (45 t), the rocket would have to be significantly bigger, ∼ 400,000 t. That is of order the mass of the Pyramid of Cheops, and might be a practical restrict for chemical rockets relating to price constraints. (emphasis added)
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Not a planetological exposition in sight so, I will add my two cents to this moderately theoretical dialogue.
Amongst exoplanetologists, the consensus has emerged that 1.6 Earth radii and 5 Earth lots is probably going to be the upper limit to rocky planets. Simulations have proven that above these figures, the our bodies develop increasingly MiniNeptune like traits. This implies very thick Helium Hydrogen atmospheres and crushing floor strain.
Additionally since Michael Hippke’s slightly whimsical paper was referenced in one of many solutions it appears applicable to say Ocean worlds at Tremendous Earth lots. Ocean worlds current a number of habitability hurdles together with a paucity of sure life crucial components like phosphorus, lack of volcanism, no water rock interface attributable to excessive strain ice on the marine ground and others. These situations will possible restrict and even forestall the institution of the colourful prebiotic chemical environments which can be needed for biogenesis.
If the primary assumption holds true, the very best gravity on a doubtlessly liveable world is not going to exceed roughly 2.5g.(edit: and thus making it not fairly so tough to succeed in orbit with chemical rockets as would have been the case with a better g worth)
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Nice solutions have been given, however one of many main themes is that they assume a set moist to dry mass ratio of 10:1 (ish). The justification is:

It’s essential to repair this as: there are not any significant solutions with with out a worth and, which worth is topic to engineering nuances, that are tough to deal with.

10:1 is an efficient choose. (We will not do significantly better than this and nonetheless have every thing work so it appears smart to stay at this)
The issue is that is the restrict of what we will make work on earth. Numerous the dry mass of a rocket is both:

straight associated to the thrusttomass ratio (i.e. quantity/measurement of engines)

not directly associated to TMR (i.e. helps the structural masses)
Observe, to maintain gravity loses equal in apply the accelerations wanted, therefore TMR, is linear with the floor gravity. Therefore so is part of the the moist/dry mass ratio.
As soon as we take that into consideration issues look lots bleaker for the excessive g superearths getting one thing into orbit utilizing chemical rockets.
The precise numbers listed here are a bit tough to know, but when 5g world results in a rocket with a w/d mass ratio of 5 to 1 (which I believe is about proper however…), you are staring down the barrel of a $10^{20}$t kind determine for launch mass. To place that into perspective, the ‘moon rocket’ is now not a superb comparability. That is the mass of the moon it acquired to.
Theoretical restrict? I might say so.
At that mass issues begin taking a flip for the ‘XKCD’. Neglect the sensible points they’re clearly lengthy gone at “moonsizedanything”. We hit chilly laborious theoretical limits.
You begin having to cope with your personal gravity.
Firstly these sensible points are huge ones even when we snicker a bit ‘engineering’ issues (like cash, and the place we’d discover $10^{19}$t of aerospace grade supplies). For instance that is the form of measurement that while you’re made out one thing strong and are already floating in house beneath 0G, you deforming beneath you personal gravity right into a ball. Attempting to make that out of principally liquid gasoline and topic it to 510g…, you are not staying the form you began. Does not matter what massratio ‘hit’ you might be prepared to take. However we have got this far, we aren’t going to let a scarcity of unobtainium cease us.
No the actual laborious restrict is how being so heavy results your exhaust velocity. On the danger of getting too meta right here when you’re heavy sufficient, its tough to get issues to return aside from you. It applies to planet sized rockets as a lot because it does planets.
Should you’re a couple of million kilos, your ‘exhaust velocity’ is the speed you will get your propellant to get to. When you have extra mass than the moon, your propellant can have misplaced quite a lot of momentum by the point it is left your gravitational affect. And that is the destiny our rocket meets.
LOX/H2 has an exhaust velocity of about $4,400ms^{1}$, about nearly as good as we will do. Let’s simply say our moonsized rocket has the density of the moon too, and so has an identical escape velocity of $2,380ms^{1}$. Then the helpful exhaust velocity of our rocket (preliminary much less escape) is lower than half. therefore half the deltav. You will not be going to house to day.
“Okay”, I hear you say, “that simply imply’s you may’t go to house in that rocket.”, “How a couple of larger one?”. Properly “No”. That is one other a kind of “Even when every thing form of labored as earlier than, you need go twice as quick, which goes to be a lot extra mass.” kind issues. Besides now we actually cannot simply take the “make in 10 orders of magnitude larger” method. Other than the truth that our rocket is now lot larger than the planet which implies we could not presumably assemble it, now we have now no likelihood of utilizing chemical rockets to propel us wherever.
To realize any momentum we have to chuck one thing out of our gravity effectively, and the exhaust velocity of chemical rockets do not make once we are this huge.
We are actually really caught.
However wait: straight the exhaust does not make it out, however I’m wondering when you may attempt totally different method of getting mass out of a really deep gravity effectively. Should not be too laborious. Even when it was solely a bit bit, we may all the time simply scale it up…
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On a sensible engineering facet of issues. In the end you might be restricted by exhaust velocity. In idea you may all the time simply make a much bigger engine, larger tanks, and so forth. Ridiculously costly, however potential. This would appear to set the actual restrict to materials energy. Materials energy is probably going to offer out earlier than the gravity wells pull exceeds the exhaust velocity of even reasonably fashionable fuels.
For instance, LF+LOX sometimes has an exhaust velocity of round 4,400 m/s. Which can battle as much as 448 G of gravity. Actually greater than the solar. Virtually nonetheless a lot lower than that. So measurement of the planet itself presents no actual deal killers, it simply makes the payloads mass fraction very VERY low.
In some unspecified time in the future although different applied sciences, like nuclear bomb drives (https://en.wikipedia.org/wiki/Project_Orion_(nuclear_propulsion)), turn into the one possible reasonably priced method off the planet.
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