Andy Weir's "Hail Mary": Question About Relativistic Rocket

[GTOM]

I am reading Andy Weir's Hail Mary.
There was a part about relativity: it wrote, that a relativistic interstellar ship had lots of spare fuel, because the mission planners actually ignored relativity, and thought travel will be Newtonian.
Am i wrong to think, that is nonsense, and the opposite is true, they should have had much more fuel, because of relativity? While the rocket experience effects of time dilation, and length contraction, but isn't the biggest effect is relativistic mass? And according to wikipedia, in the direction of acceleration, one has to calculate with CUBE of Lorentz factor.

 

[malawi_glenn]

Relativistic mass is an old and confusing concept.

The energy required to increase speed you can calculate with $\Delta \gamma mc^2$

Is your question regarding saving fuel or something else? To make a detailed calculation I think one needs to know a bit more about the actual rocket. What the thrust is etc. Usually in introductory problems one assume the acceleration phase is very short compared to the entire journey

 

[GTOM]

Relativistic mass is an old and confusing concept.

The energy required to increase speed you can calculate with $\Delta \gamma mc^2$

Is your question regarding saving fuel or something else? To make a detailed calculation I think one needs to know a bit more about the actual rocket. What the thrust is etc. Usually in introductory problems one assume the acceleration phase is very short compared to the entire journey

The question is, is it possible, that relativistic equations enable to save fuel, compared to what if near light speed space travel were only Newtonian, no relativistic effects?
The book didnt talk about saving life support (time dilation helps in that) but wrote, that a ship had lots of spare fuel, since at launch, they ignored relativity. It mentioned the effects of length contraction.
But i think the main issue is that acceleration is more and more hard near to light speed, and depend on cube of Lorentz factor.

 

[Orodruin]

The question is, is it possible, that relativistic equations enable to save fuel, compared to what if near light speed space travel were only Newtonian, no relativistic effects?

Save fuel compared to what and under what constraints? You have given no specifics regarding the scenarios to be compared.

 

[Ibix]

The question is, is it possible, that relativistic equations enable to save fuel, compared to what if near light speed space travel were only Newtonian, no relativistic effects?

I think it depends on the mission plan. For example, if your plan is "accelerate at 1g until you reach a buoy that marks half way, then turn over and decelerate at 1g" then I agree with the book. The easiest way to see it is that the ship's proper time to reach turnover is less than the Newtonian time calculation, so the engines run for less proper time than the Newtonian calculation would suggest.

If the plan is anything else I can think of off the top of my head, I think that actually you end up still moving when you reach your destination. I haven't worked out the fuel situation.

 

[GTOM]

I think it depends on the mission plan. For example, if your plan is "accelerate at 1g until you reach a buoy that marks half way, then turn over and decelerate at 1g" then I agree with the book. The easiest way to see it is that the ship's proper time to reach turnover is less than the Newtonian time calculation, so the engines run for less proper time than the Newtonian calculation would suggest.

If the plan is anything else I can think of off the top of my head, I think that actually you end up still moving when you reach your destination. I haven't worked out the fuel situation.

But isn't much more energy required to boost with constant 1g until you reach that halfway?
(The distance is several light years, maybe a dozen)

 

[GTOM]

Save fuel compared to what and under what constraints? You have given no specifics regarding the scenarios to be compared.

They want to reach another star system, then stop in it as soon as possible. They travel with near light speed. With living beings in the rocket, so no short acceleration time.

Scenario 1: they calculate with relativistic effects.
2. They dont, they think travel is purely Newtonian.

would they calculate with more necessary fuel in case of 1 or 2?

 

[Filip Larsen]

I thought about that too reading the story. For the aliens (forgot their name) to notice length contraction and being able to compensate for this in time not to overshoot (assuming they wanted to stick to their originally planned deceleration level), they would have had to continuously monitor the distance to the target star. If they just did as we humans do and precalculate the turnout time based on acceleration and time passed (i.e. without measuring the distance to the target star) then they would no doubt have either overshoot the target or discovered near the end they need to use (much) larger deceleration than anticipated.

Regarding fuel usage then one can note that relativistic travels to Andromeda using 1g can be done in a lifetime while the same Newtonian travel will take a couple of thousand years, so if you plan for a Newtonian trip and flip at the actual midpoint, then you should save potentially a lot of time and fuel. Assuming they have precalculated the midpoint to be, say, another star they pass near, then they can flip reaching this star and just be surprised that their clock show much less time than anticipated. If they periodically measure the distance to the midpoint (e.g. by noticing yet other known stars they pass) they will notice that they apparently increasingly accelerate faster and faster compared to what the ship accelerometer tells them.

 

[Dale]

They want to reach another star system, then stop in it as soon as possible.

So that mandates what we call a bang-bang control for both Newtonian physics and relativistic physics. In both cases the mission plan will be:

accelerate at 1g until you reach a buoy that marks half way, then turn over and decelerate at 1g

Or I guess you could accelerate at some different velocity, but whatever the craft’s maximum sustained acceleration is, you would accelerate at that rate first towards the destination and then away.

 

[Ibix]

But isn't much more energy required to boost with constant 1g until you reach that halfway?

Much more energy than what? I'm still not sure what acceleration profile your aliens are planning. Are they doing constant acceleration half the journey and constant deceleration the other half, or something else?

 

[Ibix]

Say the distance to the star as measured from home is $2D$ and we're going for steady acceleration for half the journey and then steady deceleration. Let's calculate how much fuel we have left at the turnover at distance $D$.

The planners use Newton. SUVAT equations tell us that we expect velocity $v=\sqrt{2Dg}$ if we accelerate at constant acceleration $g$. Plugging that into the Tsiolkovsky rocket equation we get that the mass (ship+remaining fuel) at turn over is $$m_f=m_0\exp\left(-\frac{\sqrt{2Dg}}{v_e}\right)$$where $m_0$ is the launch mass of the rocket and $v_e$ is the exhaust velocity (measured by the rocket by definition).

But the universe is not Newtonian. In a relativistic universe if you undergo constant proper acceleration $g$ then the distance travelled, $s$, velocity reached $v$, and coordinate time in the original rest frame $t$, are related by$$\begin{eqnarray*}
s&=&\frac 1g\left(\sqrt{1+\frac{g^2t^2}{c^2}}-1\right)\\
v&=&\frac{gt}{\sqrt{1+\frac{g^2t^2}{c^2}}}
\end{eqnarray*}$$which yields that the distance traveled when you reach $v$ is $s=c^2(\gamma-1)/g$, where $\gamma=1/\sqrt{1-v^2/c^2}$. Now we can look at the relativistic rocket equation, which says that $$v=c\tanh\left(\frac{v_e}{c}\ln\frac{m_0}{m_f}\right)$$The easiest thing to do is plug in our result for the planned mass ratio to get a final velocity, then plug this into the expression for $s$ and see if we have traveled further than $D$ or not. We have $$\frac{m_0}{m_f}=\exp\left(\frac{\sqrt{2Dg}}{v_e}\right)$$ so $$\begin{eqnarray*}
v&=&c\tanh\left(\frac{\sqrt{2Dg}}{c}\right)\\
\gamma&=&\cosh\left(\frac{\sqrt{2Dg}}{c}\right)
\end{eqnarray*}$$which yields that the distance traveled when the rocket has expended its planned fuel is$$s=\frac{c^2}{g}\left(\cosh\left(\frac{\sqrt{2Dg}}{c}\right)-1\right)$$If we use units of years for time and light years for distance, $c=1$ and $g\approx 1.03\mathrm{ly}/\mathrm{y}^2$. Then we can plot $s-D$ as a function of $D$:

This is clearly always positive for positive $D$. So in a relativistic universe when the rocket has used up its expected allocation of fuel it will have traveled further than the planners expect, which is another way of saying that when it's traveled half the distance to the star it'll have more fuel left than expected.

 

[Ibix]

TL;DR for the above:

Assume Newton. Use SUVAT and Tsiolkovsky to deduce the expected final mass of rocket+fuel at turnover. Plug that value into appropriate relativistic equivalents of SUVAT and Tsiolkovsky to determine the actual distance traveled with that fuel expenditure. Note that this is greater than the distance to turnover, so the actual fuel expenditure at turnover is less than expected on Newtonian assumptions.

 

[PeterDonis]

Am i wrong to think, that is nonsense, and the opposite is true, they should have had much more fuel, because of relativity?

Take a look at the relativistic rocket equations:

https://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html

Comparing them with the corresponding Newtonian equations for a particular trip (i.e., a particular distance to be traveled at a particular proper acceleration) will answer your question.

 

[jbriggs444]

But isn't much more energy required to boost with constant 1g until you reach that halfway?
(The distance is several light years, maybe a dozen)

If you thrust at a constant 1g coordinate (in the initial rest coordinates) acceleration to the halfway point then much more energy is required than the Newtonian model would predict. Indeed, it may not even be possible to thrust that long. The speed of light may intrude, preventing further coordinate acceleration.

If you thrust at a constant 1g proper (in the current spacecraft rest coordinates) acceleration to the halfway point then less energy is required because you are thrusting for a lesser proper time than the Newtonian model would predict. This results in a progressively lower coordinate acceleration and a lower peak coordinate speed.

Edit: We may be able to determine whether the speed of light limit will intrude based on the information at hand. Assume a 10 light year voyage. Halfway is 5 light years. 4.73 x 10^16 meters.

Dredge up the right Newtonian SUVAT equation: $v^2 = u^2 + 2as$ for $u$ = 0, $a = 9.8 m/s^2$, $s=4.73 \times 10^{16} m$

I make it 9.6 x 10^8 meters per second at the halfway point. This is greater than the speed of light. So there is no way to maintain a 1 g coordinate acceleration all the way to the halfway point in the relativistic model. There is, of course, no theoretical difficulty in maintaining a 1 g proper acceleration.

The predicted Newtonian average speed is 4.8 x 10^8 meters per second. Which makes the entire voyage take (2 x 4.73 x 10^16) / (4.8 x 10^8) or something a bit less than 200,000,000 seconds for the entire trip. Maybe 6.2 years per the Newtonian prediction. (Which is less than the 10 years that it would take light to get there).

Surprisingly achievable, given the availability of unobtainium engines.