# On a Super-Earth 1.5x the volume and mass of Earth, would our rocket technology allow us to reach orbit? [duplicate]

To try and make parameters clear, can we say we are talking about 50% 'more Earth'? As in, Earth, but 1.5 times as big and heavy? And let us include the atmosphere. If there is 50% more of it by volume, then that would also make it thicker at sea level, right? And then it is also affected by higher gravity?

I considered using a known Super-Earth as a reference, but the reputation of many is in question, and none seem at all promising as possible homes of intelligent aliens. Yet, the perennial question is where is everybody? Perhaps we have an unusually easy time getting to orbit from our nice liquid-water rocky planet. Maybe that is a partial answer to that question. Just a thought.

## marked as duplicate by kim holder, Machavity, uhoh, peterh, Jan DoggenApr 5 '18 at 20:24

• " Although Mercury is smaller than Mars in mass, it has a higher surface gravity since it is much smaller and more dense." Oops! Mercury is 3.70 m/s^2 and Mars is 3.71 m/s^2 so Mars still wins for 'bigger surface G'.. – Andrew Thompson Sep 13 '14 at 1:49
• @MarkAdler - Gravitational acceleration does play a role in the difficulty in reaching space. Gravitational potential $\frac \mu r$, losses due to atmospheric drag and gravity, and the rocket equation dictate how much energy is needed to put something into orbit. Gravitational acceleration $\frac \mu {r^2}$ dictate how much initial thrust is needed to put something into orbit. For example, a rocket on Earth that theoretically has ample energy to reach orbit but has an initial thrust less than $m_0 g$ won't take off. – David Hammen Sep 13 '14 at 14:18
• – kim holder Sep 13 '14 at 18:32
• Hm. I imagine putting the Earth on a photocopier and scaling it up 50%. So that would be 50% more volume, but then the increased surface gravity should increase the pressure at the surface... I don't know how that plays out or how to calculate that. – kim holder Sep 18 '14 at 14:55
• A recent paper has also appeared, discussing this a bit in a more humorous way: arxiv.org/abs/1803.11384 – AtmosphericPrisonEscape Apr 5 '18 at 4:34

I'm going to go with "1.5x the size" as referring to volume, so combined with 1.5 times the mass results in the same density as Earth.

The alternative of 1.5 times the radius or circumference combined with 1.5 times the mass would result in a Super-Earth with a sub-Earth density of 2.4 g/cc. That would be like an icy moon, not a rocky planet, and would result in a lower surface gravity than Earth (2/3 g), and the same orbital velocity as Earth, making it overall a little easier to get into orbit. That does not sound like the intention of the questioner.

So with 1.5 times the volume and mass of Earth, we would have about a 14.5% higher surface gravity, $1.5\over {1.5}^{2\over 3}$, and a 14.5% higher orbital velocity, $\sqrt{1.5\over {1.5}^{1\over 3}}$. There should be no problem at all with our current rocket technology to get into orbit from the planet, so long as its atmosphere isn't any thicker than ours. In fact, there should be no problem to stick with two stages to orbit with that modest increase in orbital velocity. The increase in surface gravity could be accommodated with a modest 14.5% increase in thrust to maintain the same thrust to weight ratio. Perhaps just somewhat larger engines or one or two more engines added to the first stage cluster.

However this Super-Earth isn't very super as Super-Earths go. Most of the discoveries are two to ten times Earth mass, as opposed to the 1.5 times Earth mass posed here.

As for an explanation for where everybody is, no, this isn't it. First off, our discoveries of predominantly Super-Earths is observational bias. It is easier to find larger planets than small ones. I'm sure that there are plenty of Earths and sub-Earths out there. Second, you can just add stages to your chemical rocket to get off of a larger planet. It will simply cost more, so you will need to be more determined. Third, you don't need to use chemical rockets. You could use electromagnetic rail launchers, nuclear fission high Isp engines, or thermonuclear bomb propulsion (see Orion). All within our current technological capability.

• I don't think nuclear pulse propulsion is much of an option for getting into orbit, at least if the planet has an atmosphere. – Rikki-Tikki-Tavi Sep 18 '14 at 10:08
• There is plenty of thrust to launch and get into orbit with the Orion concept. So long as you're not worried about a little fallout of radioactive material. – Mark Adler Sep 18 '14 at 13:57
• I was thinking the atmosphere would scale up too, so i guess that would make it 50% thicker at sea level and extending 50% further. The question isn't clear that way - i wonder if that is worth an edit. – kim holder Sep 18 '14 at 14:30
• Okay, i went ahead and changed it that way. Also the result isn't nearly as dramatic as i'd thought it would be. Yes, i could have done the math - i'm not comfortable with it yet and left it. This may lead to a follow-up question at some point, when i've figured out where the edge of possible really is for current technology. – kim holder Sep 18 '14 at 14:45
• The other option would be 1.5x the radius, density constant, for 1.5x the surface gravity, 3.375x the volume and mass, and 1.5x the orbital velocity. – Russell Borogove Sep 18 '14 at 18:03

There is a great blog post on the NASA homepage about "the tyranny of the rocket equation". It is going into quite some detail, and it gives in one of the last paragraphs:

If our planet was 50% larger in diameter, we would not be able to venture into space, at least using rockets for transport.

However, you don't want to venture in space, but reach orbit, and your Earth has 1.5 times the mass and not the diameter, so there's still hope. Just in case that the link goes dead, I am listing some of the interesting details of the blog post, as well as some additional information. The rocket equation gives:

$$M_f = 1-\frac {m_1} {m_0}=1-e^{-\Delta v\ / v_\text{e}}$$

where $M_f$ is the fraction of propellant by total mass of the spacecraft, $\Delta V$ is the necessary escape velocity and $v_e$ is the exhaust velocity. This can be rewritten to

$$\Delta v=- \ln(1-M_f) v_e$$ The orbital velocity of a body (without adjusting for aerodynamic drag) is given by

$$\Delta v=\sqrt{\frac{G M}{R}}$$

Assuming that you're interested in an Earth with $M'=1.5M$, this gives $R'=(1.5)^{\frac{1}{3}} R$, so $\Delta v'= \sqrt{\frac{1.5}{1.5^{1/3}}} \Delta v=1.145 \Delta v$, just like Mark Adler wrote in his answer.

Now we need some maximal possible values for exhaust speed and the ratio propellant/mass, which are handily given in the NASA post:

Hydrogen-oxygen is the most energetic chemical reaction known for use in a human rated rocket. Chemistry is unable to give us any more. In the 1970’s, an experimental nuclear thermal rocket engine gave an energy equivalent of 8.3 km/s. This engine used a nuclear reactor as the source of energy and hydrogen as the propellant.

And:

The common soda can, a marvel of mass production, is 94% soda and 6% can by mass. Compare that to the external tank for the Space Shuttle at 96% propellant and thus, 4% structure.

So, if we use $M_f=0.96$ and $v_e=8\;\mathrm{km/s}$, we get a maximum possible $\Delta v = 25.8\;\mathrm{km/s}$. The orbital velocity of Earth is about $7.9\;\mathrm{km/s}$, so the orbital velocity of your bigger Earth would be about $9.0\;\mathrm{km/s}$. So it would still be possible to reach orbit, also with a less efficient propellant.

• I've read that post, it's quite good. He also has a TEDx talk on the same theme. But i did mean 1.5x the volume and the mass - that's what i meant by size and mass. – kim holder Sep 18 '14 at 14:19
• I don't agree with "If our planet was 50% larger in diameter, we would not be able to venture into space, at least using rockets for transport." Assuming the same density, you would have 50% greater orbital velocity. This could be accommodated with three stages instead of two. So imagine taking one of our current rockets and making that the second and third stage. Then you'd need a really big first stage for that extra 50% in velocity and to keep the same payload mass. A very rough calculation indicates that for a 500 t Falcon 9, you would add a 1700 t first stage for 2200 t total. – Mark Adler Sep 18 '14 at 17:04
• A Saturn V was 3000 t, so a Falcon 9 equivalent payload to orbit for a 50% greater radius planet is not an unreasonable rocket to build. – Mark Adler Sep 18 '14 at 17:06

There's a very large gap between what's theoretically possible in rocketry and what's practical and what an entity has the resources and political will to accomplish.

In the staged version of the rocket equation, the possible achievable delta-V is proportional to the number of stages when mass ratio and engine efficiency is held constant. In short, you can get any delta-V you want by adding additional stages.

If you needed twice the delta-V to reach orbit (because of a larger planet and denser atmosphere, say), then instead of a 1.5 stage launcher for a one-man orbital mission (like the R-7/Vostok or Atlas-Mercury missions), a 3-stage launcher like a Saturn 1B (counting the Apollo service module as a third stage, here) might be required. (I'm handwaving a bit here; R-7 and Atlas aren't literally 1.5 stages for purposes of the rocket equation.)

Typically, each stage is 4-5 times as large as the one above it in the stack, so linear increases in delta-V requirements lead to exponential increases in overall rocket size. Complexity and cost probably increase faster than linearly with stage size, considering the assembly facilities, transport, etc. required to support them. Eventually, as you turn up the gravity, you might hit a point where a nation wouldn't be willing to take on a Saturn-Apollo sized development program to achieve the goal of a single person in orbit.

• This gets more at what i wanted to know. Of course, satellites are mighty useful, but maybe there is a point where it is cheaper to develop networks of sturdy balloons or solar-powered drones to do the communications, mapping, surveillance and so on that satellites do. Especially if there wasn't a moon to go to, it might take a very long time before people on such a planet got out into space. – kim holder Sep 18 '14 at 17:51
• In the intermediate cases, you'd just put a lot more effort into miniaturizing your satellites. Vanguard 1 launched less than 6 months after Sputnik, and accomplished a lot more, while weighing 1/50th as much. – Russell Borogove Nov 4 '14 at 21:34