Interstellar story-telling that is aiming for “Hard-SF” credibility is a natural subject matter for Crowlspace. So how does Andy Weir’s latest “Project Hail Mary” rate?
First, a quick precis, which will contain spoilers.
Man wakes up in a room with a robotic system that is clearly keeping him alive. How did he get there? And where is ‘there’? Through a series of flashbacks the protagonist – a radical astrobiologist turned Science Teacher – remembers he is on a mission to another star system and to survive the journey his crew were put into induced comas for several years, but only he survived the coma, with a damaged memory. At the same time he conducts a series of basic physics experiments to deduce his situation – he’s in a starship decelerating at 1.5 gee, with just days remaining before arriving at Tau Ceti.
The means for getting to Tau Ceti at 1.5 gee is also the reason for going – a space-living organism that perfectly interconverts mass-into-energy and back again is feeding on the Sun and other nearby stars, but not Tau Ceti. To save the Earth the “Hail Mary” has been sent to Tau Ceti to find out why the Astrophage isn’t eating Tau Ceti, despite originating there. By encasing the crew capsule in Astrophage, the radiation issue of ultra-relativistic travel is solved, as well as the energy handling issues in the “spin drives” that are perfect Photon Drives enabled by Astrophage’s matter-energy handling.
Once there the protagonist encounters the sole survivor of a similar mission sent by the inhabitants of the Super-Earth that orbits 40 Eridani. So, of course, the alien is dubbed an ‘Eridian’ by the human. The alien lives in a habitat at of ammonia gas at 29 atmospheres pressure and 210 C, with an integument mostly made of exuded metals, though with a squishy organic core within. The two learn to communicate despite the alien being “blind” (they ‘see’ via natural sonar) and the human being challenged by the Eridian’s heat output. Working together they solve the mystery…
A Crowlspace analysis always starts with the propulsion system and the interstellar trajectory. The time to get to Tau Ceti (11.9 light-years away) is roughly 13 years and the ship-board time is vaguely 3-4 years – all at 1.5 gee. The maximum speed that the Astrophage naturally attains is 0.92 cee, as measured by red-shift/blue-shift of its radiating frequency in the infra-red. Each cell of Astrophage has a rest mass of ~0.5 nanograms and can store 17 nanograms of mass-energy, implying a mass-ratio of 35. The “Hail Mary” carries 2,000 tons of energized Astrophage and has a dry mass of 100 tons, thus a mass-ratio of 21.
The Mass-Ratio (Mo: initial Mass; Mf: final Mass) for a perfect Photon Rocket which brakes to a halt from a maximum speed (B = v/c) is computed by:
Mo/Mf = (1+B)/(1-B)
To just attain B the mass ratio is computed by:
Mo/Mf = SQRT[(1+B)/(1-B)]
As an example, the mass-ratio for a B = 0.92 c is (1.92)/(0.08) = 24
So you can immediately see some questions arise in my mind about the performance figures implied.
What about that trajectory?
For a given displacement, S, the time to travel a continuous acceleration (Brachistochrone) trajectory at acceleration, g, is:
t = SQRT[ 4*S/g + S^2 ]
Onboard the vehicle the time dilation will be significant, giving a Tau time of:
T = 2*(c/g)*ACOSH(1 + g*S/2.c^2)
Plugging in 11.9 light-years at 1.5 gee, we get t = 13.13 years and T = 3.894 years.
Note: according IAU definition one light-year is the distance travelled by light at 299,792,458 m/s in 31,557,600 seconds (8,766 hours or 365.25 days, an average Julian Calendar Year). Some old definitions used the Mean Tropical Year of 1900, which was 31,556,925.9747 seconds.
Just how fast is the “Hail Mary” travelling at its peak in that case?
If a rocket measures its speed by an integrating accelerometer – it multiplies the acceleration measurements per unit time and then adds them up – then exceeding the Speed of Light is entirely doable. The “Hail Mary” accelerates for a total of 3.894 years, so half that time it is speeding up and the rest it is slowing down. Therefore how fast is it doing at the mid-point?
U_max = g*T/2 = 1.5 gee x 3.894 years x 0.5 x 1.0323 = 3.015 cee
To convert into the relativistic speed B in the rest frame of the wider Universe:
B = TANH(U_max)
Inputting the current data we get B = 0.9952 cee
Therefore the required mass-ratio is:
Mo/Mf = 1.9952/(1-0.9952) = 415.6
Which raises the question in my mind: what did Andy Weir miss when doing his calculation?
I suspect he has confused the mass-ratio for getting to speed B for the mass-ratio for speeding-up/slowing-down again. A mass-ratio of 20.4 will get a vehicle to 0.9952 c, but not be able to slow it down again.