# Launch-to-orbit vs. planetary mass question

#### ewflyer

Hello all,

My name's Greg. I'm an airline pilot (my day job) and a fiction writer (my hobby). Right now I'm working on a story that borders on science fiction (a first for me).

I've got a question: What is the upper limit on planet size (mass) that would effectively rule out putting satellites into orbit? (Given our current level of technology, materials available, and all the standard orbital considerations such as the planet's atmosphere and the lateral acceleration needed to remain in orbit)

It would seem that the Earth is very conveniently sized for us to put things into orbit. The Earth is (obviously) perfectly sized and situated for life (us) without being so massive as to prevent the launch of satellites, space missions, and space probes.

But if the planet were more massive wouldn't there would be a point beyond which you just couldn't construct a lift-vehicle to get to orbit? I mean, assuming you're working with the same Periodic Table of the Elements in your rocket fuel composition (no "unobtanium" types fuel or metal alloys allowed, or "nutty professor" technology for that matter).

Is this a subject that's been discussed in scientific circles? Is there information out there that touches on this? If anyone knows of sources I'd love to know about them.

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#### JaredJames

Well strictly speaking, it's only when the escape velocity > c that you cannot leave said planet, but there would be a practical limit so far as available energy for launch goes. But that is determined by what energy production facilities you have for rocket (or other) propulsion systems.

#### alexjbuck

Alright,

The basic method I used was to apply the ideal rocket equation which relates a delta V (velocity change) to a mass fraction, with the exhaust velocity considered.

the way I approached this was by saying that any fictional planet will have a similar composition to the earths, thus its density is roughly the same. Using this you can find the planet radius as a function of its mass.

density=(6e24 kg)/(4/3*pi*(6378000 m)^3)
planet volume (Vp) =planet mass/density

now, the circular velocity of an orbit is sqrt(mu/r) where mu is the product of G (gravitational constant, 6.67e-11) and the planet mass and r is the orbit radius (we will make this an altitude of 200km, thus the planet radius + 200km, this is standard for a Low Earth Orbit).

the ideal rocket equation states that
deltaV = Vexhaust * ln(initial mass / final mass)

On earth, circular velocity at LEO is ~8km/s and a rough value for total deltaV to orbit is ~10km/s. Using this relation of ~1.25*Vcirc = deltaVlaunch we can generate the required deltaV's to launch on these fictitious planets based off the previously calculated circular orbit velocities.

Finally, using a nominal value of 4000 m/s as exhaust velocity I arrived at the following relation between planet mass and (final mass / initial mass) ratio. Now this is for a single stage to orbit analysis. As you can see even for 1 earth mass the mass ratio is only ~.07 meaning 93% of your initial weight is fuel. We don't launch this way on earth, we use staging. Staging allows you to not have to lift the weight of the fuel tanks etc for the fuel that is consumed.

The second and third lines on the above plot represent a 2 stage and a 3 stage vehicle where at each stage separation the mass ratios are m_final_previous_stage/m_initial_next_stage = 1.5, essentially dropping 1/3 of the current weight off (the fuel tanks etc for the spent stage).

This comes out to a simple ratio multiplication problem.

m3_final/m3_initial * m2_final/m2_initial * m1_final/m1_initial = total mass ratio for the rocket. If no staging occurs then m3_initial = m2_final and m2_initial = m1_final.

The way we get an advantage is by causing m3_initial to NOT equal m2_final by dropping off mass. In that case the final mass ratio becomes,

m3_final / m1_initial * m2_final/m3_initial * m1_final/m2_initial.

So a reasonable value for the m2_final/m3_initial and m1_final/m2_initial terms is 1.5 thus for a 3 stage rocket the final mass ratio (m3_final/m1_initial) is just 2.25 times the mass ratio for no staging (m3_final = m1_final if there's no staging).

SO, finally answering your question. From the plot I would venture that a technologically sophisticated civilization might be able to reach orbit with a planet twice the mass of earth, conceivably upwards of 9 times the mass of earth if they are willing to accept .01 mass ratios (99kg of fuel for every 1kg of stuff reaching orbit, including the structure!).

This is all assuming chemical propellants (LOX/LH2) However if they are really good at building rockets and/or can build nuclear thermal rockets its possible the exit velocities could be significantly higher. A doubling of exit velocity would come through as a square root of the mass ratio. Since our mass ratios are ~.1 this is actually an increase in the mass ratio. With this assumption its concievable that planets upwards of 8 earth masses could have a .1 mass ratio and planets of up to 50 earth masses could have a .01 mass ratio. I'm not including the plot for this step since its just a square root of the values on the shown plot.

questions and criticisms both are more than welcome

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#### ewflyer

SO, finally answering your question. From the plot I would venture that a technologically sophisticated civilization might be able to reach orbit with a planet twice the mass of earth, conceivably upwards of 9 times the mass of earth if they are willing to accept .01 mass ratios (99kg of fuel for every 1kg of stuff reaching orbit, including the structure!).
Thanks.

My interpretation of your data is that as you increase anywhere beyond about twice the mass of our Earth, assuming the same technology we're currently using, the launch to orbit mass-ratio reaches a point that would probably be cost prohibitive to any single planetary society (like the United States, for example).

I mean that in a society such as ours, but with a twice as massive planet, the "return on investment" for all the resources you'd need to launch satillites or other missions (like our space shuttle or Apollo) at anywhere near the same rate probably couldn't be justified to the public. On this hypothetical planet the NASA budget would be a national back-breaker.