How a lot larger may Earth be earlier than rockets would not work?

Question

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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 1.1³, 1.2³, 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 cut-off, 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!

requested Mar 9, 2016 at 7:45

uhohuhoh

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31

Accepted Answer

As a result of linear will increase in delta-v require exponential will increase in mass, small adjustments to the assumptions you make about gasoline tank structural mass and engine thrust-to-weight 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 7-stage 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 delta-v 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 Apollo-era rocket engines: RL-10, J-2, M-1, H-1, and F-1.
First-stage TWR at ignition should be a minimum of 1.2 (relative to native gravity)
Center-stage TWR at ignition should be a minimum of 0.8
Ultimate-stage TWR at ignition should be a minimum of 0.5

Given these assumptions, here’s a desk of floor gravity, stage depend, first-stage engines, and whole rocket mass.

Floor                         First        Complete       Saturn V 
Gravity   Levels                Stage      Mass, t     Equal
 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      hundreds of thousands
10.0          18        2.88e19   F-1      1.65e21  quadrillions

Up above 10g, one thing actually attention-grabbing 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 super-Earth-based 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 thrust-to-weight and thrust-to-cost 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 first-stage 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.