4
$\begingroup$

So, I read this article: https://techcrunch.com/2018/02/22/spinlaunch-2/

(TechCrunch seems to have issues with this URL: here's one from the Wayback Machine that seems reliable: https://web.archive.org/web/20180228071731/https://techcrunch.com/2018/02/22/spinlaunch/)

If I understand correctly, the SpinLaunch folks plan to launch vehicles into space by spinning them in a centrifuge and then releasing them, presumably at or above escape velocity, along a path tangential to the centrifuge.

This didn't feel right to me, so I looked up some information for starters:

  • Per Wikipedia, humans can withstand up to 20G for a few seconds in the right conditions.
  • Centripetal acceleration is v^2/r
  • Earth's escape velocity from the surface is ~11.2km/sec

Then I did some math:

Solve for r:
eqn

I come up with r ~= 639543 meters.

That is one big centrifuge. So, let's go ahead and assume that we're not putting humans into space with this thing, and dial things back a bit. Let's say the centrifuge would fit in the world's most voluminous building: Boeing's final assembly site in Everett, WA.

To my eye, the building seems to be about 2000 feet on a side, or 600 meters. (I tried Googling for actual dimensions; only found figures for total volume and acreage.) Now we're looking at ~21,000G of centripetal acceleration.

When I came up with that number it reminded me of numbers I've seen for instantaneous acceleration during high-speed car accidents and other such high-impact situations. I did some googling to look into material deformation due to acceleration, and didn't find much that wasn't dealing with mechanical impact or was waaaay over my head, but intuitively I think that the sorts of things we like to put in space (e.g. satellites) aren't sturdy enough to survive such a launch.

(I also considered CERN as a point of reference for the size of the centrifuge. It's 27.6km in diameter, which yields centripetal acceleration of 479G. Potentially more manageable from a material standpoint for cargo, but massively less practical from the standpoints of construction, space constraints, launch mechanism materials, etc.)

In both cases, I'm extremely skeptical about the feasibility of the amount of energy that would be required to spin up such a centrifuge.

So, here I am, asking you kind physicists and physics enthusiasts: please fact-check my intuition. Am I crazy, or is (my interpretation of) SpinLaunch's plan impossible? Do any of you see a way that a centrifuge-based space launch system could be brought into the realm of the practical?

$\endgroup$
  • 1
    $\begingroup$ My quick opinion based on reading the article and looking over your numbers is that this launch idea is pretty questionable, too. The g-forces required to get to the required escape speed without building a gigantically large wheel will be huge, and there are all kinds of questions about air drag, the mechanical strength of the wheel, and the required amount of power. Also, the 11.2 km/s escape velocity ignores air drag. Much higher speeds would actually be required. Don't even know if much of a vehicle would survive suddenly being launched up into air at +11.2 km/s or if it would all burn up. $\endgroup$ – Samuel Weir Feb 28 '18 at 23:01
  • 1
    $\begingroup$ Reminds me of this old project: "Super High Altitude Research Project (SHARP)": en.wikipedia.org/wiki/Super_High_Altitude_Research_Project $\endgroup$ – Samuel Weir Feb 28 '18 at 23:16
  • 2
    $\begingroup$ Not only do you need to consider $g$ but also the rate of change of $g$ when the vehicle is released. The sudden jerk can also be hazardous. $\endgroup$ – sammy gerbil Mar 1 '18 at 9:15
  • $\begingroup$ I thought about the jerk too, but I didn't think it would reach hundreds, let alone tens of thousands of Gs, so I focused on the first problem. :) $\endgroup$ – JakeRobb Mar 2 '18 at 21:46
  • $\begingroup$ @SamuelWeir thanks for pointing out air drag. I googled for it quickly and naively assumed that it had been accounted for. But of course, you can't do that because you don't know the Cd or frontal area of the object doing the escaping. $\endgroup$ – JakeRobb Mar 2 '18 at 21:47
1
$\begingroup$

I got interested in this and went deep. Here's my desk-engineered "feasibility study" for a simple centrifuge in a giant vacuum chamber. I.e. a huge rotational bearing in the middle, spinning a long arm that releases the projectile at a precise time (so it exits through a plasma window or thin rupture disk).

Estimating the scale

I'll choose a 500kg projectile, which is the upper end of the "microsat" payload size. And SpinLaunch's advertised 13300 m/s launch speed.

Centripetal force exerted by the end of the centrifuge arm will be $mv^2/r$. So the required forces get smaller as you increase centrifuge arm length. So one question is, how big can you build a circular vacuum chamber, with no internal supports? Suspension bridges, for example, hold up more than 14psi and can extend well past 1000m, so you could theoretically go pretty huge. But I'd guess a vacuum chamber with radius 100 meters is probably near the limit of what you could build with a few tens of millions of dollars. So with R=100m the tip of the arm needs to exert $mv^2/r$ = 8.9e8 Newtons on the projectile (an extreme amount of force!).

Will it hold together?

You also need the centrifuge arm to not fly apart from the centrifugal force on its own mass. So you need a high strength-to-weight ratio material; carbon fibers are at the top of currently available materials. Microscopic structures like carbon nanotubes and colossal carbon tubes have 20X better performance, but, as far as I know, can't be made at macroscopic size with current technology.

The centrifuge arm's mass turns out to be significant. It yields the same problem as space elevators: the part at the very tip needs to be thick enough to support the load. The next part needs to be thicker, to support the tip AND the load. The next bit needs to be thicker still, etc. If the material isn't strong enough, the required thickness becomes so vast that you could never build it. I made a Python script to do the numerical integration to determine arm cross-sectional area:

from matplotlib import pyplot
import numpy as np
for R in [50, 100, 150, 200, 300]: #centrifuge arm radius
    vehicle_mass = 500 #kg (arbitrary choice)
    v=13300 #m/s (as advertised)

    GRAMS_PER_CM3 = 1e-3 / (.01 ** 3)  # factor to convert g/cm^3 to kg / m^3

    INVENT_NEW_MATERIAL = True
    if INVENT_NEW_MATERIAL:
        #colossal carbon tubes, if they could be made macroscopic (according to the Wikipedia article above)
        material_name = 'Colossal carbon tubes'
        arm_density = 0.116 * GRAMS_PER_CM3
        arm_tensile_strength = 6900e6  # Pascals (Newtons tension per m^2 cross-sectional area)
        arm_derate_factor = 0.8  #thin safety margin for aerospace
    else:
        # numbers for the top carbon fiber listed at https://en.wikipedia.org/wiki/Specific_strength
        material_name = 'Existing carbon fiber'
        arm_density = 1.79 * GRAMS_PER_CM3
        arm_tensile_strength = 7000e6 # Pascals (Newtons tension per m^2 cross-sectional area)
        arm_derate_factor = 0.8 #thin safety margin for aerospace


    def requiredCrossSectionalAreaFor(tension):
        return tension / (arm_tensile_strength * arm_derate_factor)

    omega = v/R

    dL = 0.005 #differential length for my numerical integration
    def centripetal_acceleration_at(r):
        return omega**2 * r

    numFiniteElements = int(round(R/dL))
    radii = np.cumsum(np.ones(numFiniteElements)*dL) #radius at finite element i. E.g. dL, 2dL, 3dL, etc.
    crossSections = np.zeros(numFiniteElements) #will hold cross-sectional area of arm, as we go out from the central pivot
    tensions = np.zeros(numFiniteElements) #will hold tensions as we go out from the central pivot
    tensions[-1] = vehicle_mass * centripetal_acceleration_at(r=R) #tension in outer tip of the arm
    for i in reversed(range(0,numFiniteElements)):
        crossSections[i] = requiredCrossSectionalAreaFor(tension=tensions[i])
        if i > 0:
            myMass = crossSections[i] * dL * arm_density
            forceToHoldMe = myMass * centripetal_acceleration_at(r=radii[i])
            tensions[i-1] = tensions[i] + forceToHoldMe

    pyplot.plot(radii, crossSections, label="R=%.0fm"%(R))
pyplot.title("Centrifuge arm cross-sectional areas. \nmaterial: %s\nV=%.2fkm/s, vehicle M=%.0fkg"%(material_name,v/1000.0,vehicle_mass))
pyplot.ylabel("arm cross-sectional area ($m^2$)")
pyplot.xlabel("position along arm")
pyplot.legend(loc='best')

The verdict

With space-elevator technology (macroscale colossal carbon tubes or nanotubes) the arm could hold together. A 100m arm would need a cross section of 1 square meter at the thickest point. Unobtainium looks like it could work

With existing technology (carbon fiber) the arm would need a cross-sectional area of billions of square meters. So, IF we assume a simple centrifuge with current materials, the myth is busted! Carbon fiber, no chance

Caveats

I've assumed a standard centrifuge, but that might not be SpinLaunch's plan. They might be thinking of something like a circular Hyperloop, where the projectile presses against the outer wall. Accomplishing maglev at these forces and at orbital speed seems like its own insanely hard problem, but I'm not prepared to prove that it's infeasible.

They might also be doing some clever dynamic physics, like a giant whip crack effect. Or a centrifuge carrying another centrifuge. It's hard to see those working, but maybe. Or possibly the advertised 13km/s was just bluster, and they're actually planning a lower exit speed plus a rocket booster. Or possibly they've got nothing.

If it worked, could it get through the atmosphere?

Gun launch to space was evaluated by the DoD as part of SDI in the 80's and early 90's (they focused on railguns and coilguns). If you read their initial studies, they thought the basic concept was feasible: if your projectile is a long, thin rod with an ablative nosecone, it can punch through the atmosphere and still be at orbital speed. It's basically the same idea as the bunker-busting bombs that can punch through many meters of concrete. You wouldn't put humans in it, but it could carry tanks of fuel or raw materials.

$\endgroup$
  • $\begingroup$ Thanks! I hadn't considered the feasibility from the standpoint of the feasibility of building the centrifuge itself. Still super curious about the energy it would require to spin the thing up. $\endgroup$ – JakeRobb Aug 14 at 16:11
  • 1
    $\begingroup$ The energy wouldn't be a big problem actually; if we consider just the 500kg vehicle then 0.5mv^2 is 44e9 joules, which, if I did the math right, is about $1228 of electricity at 10 cents/kWh. The centrifuge arm could do regenerative braking when it slows back down after the launch, but even so, inefficiencies would probably put you at 10-100X that ideal energy use. (Still crazy good compared to rockets, which is why non-rocket spacelaunch is such an attractive concept.) $\endgroup$ – Luke Aug 16 at 8:28

protected by Emilio Pisanty Jan 27 at 22:06

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?

Not the answer you're looking for? Browse other questions tagged or ask your own question.