Pettit's handwave

Proxima's planet, as Earth sees how it redshifts and blueshifts its star, is 1.3 our mass. And since its visible orbit is not a transit, that 1.3 is a minimum; the thing could be rolling along at an angle to us. If you land humans on Proxima's planet, they will struggle to get off again. Worse than we did from Earth.

Don Pettit (pardon the misspelt URL) did the Tsiolkovsky equations (like a sucker), where m_init = 1 and m_final = 0.04.... or claimed he did. Pettit concluded that a planet 1.5 times the diameter(!) of ours could not get a rocket out of the gravitational well. How did he get here?

Proxima is a decent proxy for that hypothetic Big Earth. We can agree that a solid planet heavier than Earth should be larger as well (and that a less-dense planet, like Venus let alone an ice/water world, will be). We can agree that this star is likely forcing Mercury-slow rotation upon a planet so close, such as not to be forcing an oblate shape. We can also agree that the atmosphere on Proxima is unlikely to be Venus-thick nor consistently Venus-fast: so we cannot count on a high-flying spaceport. I don't know about equatorial mountains, but we can agree to neglect them. Sea-level G is the G what Proxima's planet gives us.

m_final is, of course, the payload's ratio so the highest Pettit allowed with a chemical rocket (nuclear might allow higher m_final). ln(m_inf / m_final) comes to 3.219 - as minimum (nuclear m_final may yield lower ratio, here). So delta V >= 3.219 exhaust V depending on the rocket's efficiency.

My next challenge was how Pettit's maths got him to radius being the magic quantity; rather than, oh, mass. Stack Exchange scratched its head too.

But eventually I realized - Pettit is talking escape V, and has swapped escape V for a function of surface-gravity and radius. He maintains 9.8 g to allow his alt-Earthers a humanlike form. Pettit assumes his larger (and heavier) planet is less dense than 5260 kgm^-3. To be exact, his Big Earth is 3624 kgm^-3: which looks to me barely more than pure rock and charcoal density under internal gravitational pressure. Let's pretend that Proxima's inner planets are indeed this iron-poor. (A thick layer of ice and/or water over a denser core would also do it, if we weren't talking Proxima.) Maybe the titanium for the rockets drop in from an interstellar comet. BUT ANYWAY

So the delta V = Math.Sqrt(2g) * Math.Sqrt(R), also a minimum; it is this which must >= 3.219 exhaust V. Assuming g stays 9.8 on prairie, adding planetary radius does indeed add to the escape velocity. So: exhaust V <= 6.093 * root(radius). Backtracking from 9680 km, Pettit refuses exhaust-V above 18957 ms-1. I'm not bothering with the maths anymore; maximum specific impulse horizontally would be 18957 / 9.807 = 1933 seconds, but our rocket is going up and trimming g down. Perhaps there exists no singlestage propellant to accelerate to that speed before the gas runs out.

The Stackers first noted that multiple stage rockets could surmount this. Also, the consensus arrived that Pettit's whole exercise is moot. Nobody serious cares about leaving the well on the first go. They just want into orbit.

Once in orbit (Proxima's inner planet probably doesn't have its own moon) Prox-Simians are Halfway To Anywhere, and can start assembling a depot. (Assuming they haven't already inherited one from the first colonists.) The planet can meanwhile use planetside mass-drivers to fling nonfragile materiel up there - starting with the fuel needed to get back down. The escape velocity from that orbital depot is what matters now.

So Pettit was wrong, and Ross Pomeroy did not follow the math. Pomeroy needs to do more of that if he's pretending to be a journalist on this field. As for Pettit, he needs to show his work better.