Physics Shower Thoughts Part II: Cosmic Radiation Pressure and Relativistic Spacecraft

This is the second instalment of "Physics Shower Thoughts," an exploration of an idea that I found interesting that ultimately may not have monumental consequences in the grand scheme of our understanding of the universe. In this case I was inspired by a question on reddit that was answered by another physics blogger, VeryLittle of Quarks and Coffee. The question was about the limits of acceleration given special relativity, and my shower thought was about the role of blue-shifted radiation pressure in limiting that acceleration.

The setup is thus: an object (let's call it a spaceship) under the influence of a constant force in its reference frame will accelerate. The acceleration will seem uniform but as it gets very fast relative to its initial rest frame, its velocity will plateau asymptotically towards light speed. In our universe, however, there is a cosmic microwave background that will be blueshifted in front of us as we move in one direction, and redshifted behind us. The radiation pressure from this background will become stronger and counter the force that our spaceship is producing. This may limit the ability of our spaceship to accelerate. So, to what extent does this matter?

It's a good picture, ok.

To figure this out, we need to work out two things: the blue-shifted blackbody radiation pressure, and the relativistic acceleration under special relativity. The shifting of the blackbody spectrum can be considered with varying degrees of complexity, including a Lorentz transformation of the entire spectrum and a mixing of polarizations. However, I am interested in a simple solution and an order-of-magnitude estimate, so I will be using a simpler method: I will consider the Stefan-Boltzmann law given a relativistic Doppler shift of the peak frequency of the blackbody distribution.

Incident electromagnetic power can be converted to a radiation force by dividing by the speed of light. The Stefan-Boltzmann law gives us areal power density, so we can multiply by the cross-sectional area as well to from power density to pressure to force:

$F_{r}=\frac{A}{c}\sigma T^4$

The Stefan-Boltzmann constant $\sigma$ is a product of several powers of Boltzmann and Planck's constants as well as the speed of light, but its value in SI units is easy to remember: 5-6-7-8, or 5.67x10$^-8$ Watts per meter squared per kelvin fourthed.

According to Wien's displacement law, the temperature is proportional to the peak frequency of the distribution, which I'll call f. In our universe at this time, f is about 160 GHz. The proportionality constant is based on numerically minimizing the Planck spectrum, and in terms of frequency Wien's law can be simply expressed:

$T=\alpha f$

The proportionality constant for frequency is actually very close to two times the constants of Pythagoras* and Boltzmann divided by that of Planck, or in SI units about 5.9x10$^{10}$ Hertz per kelvin.

Moving toward a source, the frequency experiences a Doppler shift and is transformed into f', and if the motion is fast enough we should take into account the full relativistic Doppler shift:

$f'(v)=f\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}$

The radiation hitting us from behind is redshifted, and I will deem it small enough to ignore. Plugging this all together, the transformed cosmic radiative force on our spaceship is:

Now we consider the relativistic acceleration of this spacecraft. The way to incorporate special relativity into Newton's law of motion is to remember that the change in momentum is the product of force and time. Momentum is the product of mass, velocity, and the Lorentz factor. If we just have a constant force then we can find the acceleration as a function of time:

If you plot this vs time for some values of F and m you will find that it increases and asymptotically approaches light speed. Somewhat coincidentally, if you set F/m to be Earth's gravity, it will take about a year to get close to light speed.

Accelerating under Earth-gravity-equivalent for three years. To convert seconds to years, remember that pi seconds is a nanocentury.

If we then incorporate our radiation pressure into the force side of the equation, it gets a bit more complicated.

This does not have a closed form expression for the velocity, but if I solve it numerically I find.... that it increases and asymptotically approaches light speed. For the values I use, it is essentially indistinguishable from the radiation-free solution. However, if I calculate values for long enough times with high enough numerical precision, the radiation solution does converge on a sliiiiiightly subluminal value (99 point a bunch of 9's percent light speed). So, what's going on?

This is going on.

Basically when the spaceship is going fast enough, the radiation force becomes strong enough to counter the accelerating force, and the net force is zero. This velocity is easier to calculate, we just solve the transformed radiation pressure equal to constant F, and solve for v. To simplify notation, I have introduced the variable pressure P=F/A:

Just the general form of this expression is $(1+x-2\sqrt{x})/(1-x)$, which is monotone decreasing, passing zero when x=1. In our scenario, that corresponds to $f^{4}/P=\alpha^{4}c/\sigma$, which I guess is the frequency and pressure required to have zero acceleration from the rest frame.

So, let's plug in some realistic values for this critical velocity and see what happens. Using the space shuttle as a framework for a spaceship, it has a mass of two million kg, a thrust of 12.5 meganewtons, a cross-sectional area of roughly 200 square meters. This puts its maximum velocity at 99.99999992% the speed of light, given constant thrust.

Now there are many other factors I didn't take into account. I could use a more rigorous transformation of the blackbody radiation. The length-contraction of the spaceship might change the incidence of the radiation, compounding the effect (it's kind of funny that shape becomes important again, not for aerodynamics but for electrodynamics). The mass could decreases as fuel is ejected from the rocket (maintain fuel supply for this journey could perhaps be attained with a Bussard ramjet, but that's beyond the scope of this article), which increases the acceleration. And it would take so long to accelerate to these high speeds that the cosmic background radiation could become more significantly redshifted due to the intervening cosmic expansion.

As VeryLittle pointed out in his reddit post, such a cosmic speed limit already exists for protons: the GZK limit is the speed at which cosmic ray protons see the CMB blueshifted to the point that it interacts with the now-gamma rays. This corresponds to an energy of 8 joules, but the speed is close enough to light that the number of 9's in the percentage doesn't really matter (whereas for the space shuttle it's only 7 decimal 9's). I guess protons are more electroaerodynamic than spaceships.

So, to summarize, the blueshifting of the cosmic microwave background may have an extremely tiny effect on the limiting speed of relativistic spacecraft. However, if I were a relativistic spacecraft engineering, I wouldn't worry about it.

*also known as the square root of two.