Just in case you happened to arrive at this page inadvertently, a few words of introduction may avert an in-spiral into a perplexity singularity. This document provides the technical background for my science fiction story Trek's End. While there are no real “spoilers” here, unless you've read the story you'll probably be baffled why anybody would fret over the motley collection of details discussed herein. So, if you've not done so already, go read the story, and I'll see you back here later.
OK, onward into the twisty maze of little calculations, all highly approximate…. While Trek's End is the antithesis of “hard” science fiction, nevertheless one likes to get the details right. This entails some actual computations, which I'll sketch out for you. The notes are in the order they come up in the story, identified by the phrase they apply to, which is cited in italics.
Doubt this is possible? It's easy for an adult male to exert 4 kg of force purely by squeezing the fingers of one hand together. This is actual science—I tried it on the bathroom scale! In fact, it's pretty easy to get close to 3 kg with an index finger alone. I weigh about 75 kg, so that works out to an acceleration of 0.067 m/s². The velocity from the finger push-off is limited by the maximum distance the fingers can remain in contact with the wall, say 6 cm (0.06 m). You won't be able to maintain the maximum force over the entire push-off, so let's take 3 kg as an average, which reduces the mean acceleration to 0.04 m/s². From Newton's second law and a little calculus, we find you'll be able to exert this force for about 1.7 seconds before your fingers lose contact with the wall. After this, trivially, your velocity will be 1.7×0.04=0.068 m/s, or about 7 centimetres per second, which is about as fast as I'd want to go zipping around a small room in zero G. Obviously, pushing off with both hands doubles that speed. Flex your elbows and really push, and you may arrive at the other wall too quickly for a graceful landing. A smaller person, of course, may not be able to exert the same force but won't weigh as much either.
Without giving the distance from which one is viewing the soccer ball, this is really nonsense, but that's how most people describe angular extent—by naming an object viewed from the usual range at which they encounter it. Here, Pete (actually, I haven't given his name yet in the story) is using this shorthand for a soccer ball viewed at around arm's length. Geosynchronous orbit (GEO) is 35,785 km above the centre of the Earth. The Earth's equatorial radius is 6378 km, so viewed from GEO the Earth subtends 2×asin(6378/35785), about 20 degrees. At arm's length, about 60 cm, a sphere of about 21 cm diameter has the same angular extent. (You can work it out using similar triangles without ever touching a trig function: 2×(60×(6378/35785))=~21.38. The factors of two are because we're using the Earth's radius, the figure usually cited, rather than its diameter.) Soccer balls come in different sizes (I never knew that before, did you?); the “size 5” used by adults is about 22 cm in diameter—close enough. Humans are so notoriously poor at estimating angular extent that even the difference in observing a soccer ball near your feet wouldn't make much difference in the perception. Have we now sufficiently beaten this one to death?
The court is cylindrical, with a net stretched across the middle spanning about 1/4 of the diameter of the cylinder. Teams spread out in space on either side. It's permitted to pitch the ball on either side of the net, or bounce it off the walls of the cylinder. One team scores when the other allows the ball to strike the end of the cylinder they're defending. Players can wear “wings” on their arms and legs to “fly” through the air, and/or launch themselves off the walls. Actually, the game would be more like squash or handball, but I think you'd need something that moves more slowly like a volleyball given the constraints of maneuvering in microgravity.
“L/D” is pilot-speak for “lift to drag ratio”, the distance an unpowered aircraft travels forward for each unit of distance it descends. Gliders have very high L/D, while watermelons do not, and are thus prone to sloppy landings.
Joe Costello, then CEO of Cadence Design Systems, once said, “I've never met a human being who would want to read 17,000 pages of documentation, and if there was, I'd kill him to get him out of the gene pool.”
The parameters of the impactor need to be carefully tuned to be consistent with the reported observations, consequences of impact, and requirements of the balance of the story. I originally did the algebra and hand-calculated some cases to see if I could make it all work. After I was convinced I could, I wrote a pair of Perl programs to help me fine tune the parameters for the story. You can download a Zipped archive containing these programs if you'd like to play around with them yourself.
The starting point is the energy required to boil off all the oceans and sterilise the biosphere to a depth of about 1 kilometre. This is estimated as about 1035 ergs, or about 2×1012 megatons of TNT. (By convention, a megaton is defined as 4.184×1022 erg.) Just to be “safe”, since the extreme velocity collision I envisioned will radiate a substantial part of its incident energy back into space, I chose an impactor energy of 1036 ergs—ten times the minimum requirement.
The kinetic energy of an impactor is given by the equation e=½mv² where m is the mass in kilograms, v the velocity in metres per second, and e the energy in Joules. To convert Joules to ergs, multiply by 107. Since the relativistic correction to this equation goes as v²/c², it may be neglected at the velocity I intended.
While one can obtain any given energy by choosing m and v which yield it (as long as v is less than the speed of light!), I also had to ensure the approach of the impactor could be plausibly observed by a human from geosynchronous orbit. Thus, it had to reflect sufficient sunlight so as to be bright enough to be seen. The apparent magnitude (visual brightness) of an object illuminated by the Sun depends on how far it is from the Sun and the observer, how big the object is, what percentage of sunlight it reflects (its albedo), and the fraction of its disc (assuming it's spherical) illuminated by the Sun. The program magnitude.pl, included in the download archive, performs this calculation.
In the story, the pilot notices the approaching object after having looked up from a computer screen to gaze at the Earth. Consequently, his eyes will not be dark adapted and, in any case, will be dazzled by the mostly illuminated disc of the Earth. It's unlikely that even the brightest stars, which are about magnitude −1 (lower numbers are brighter) would be visible in these circumstances. Having chosen a time for the event when the Earth would be mostly illuminated (so the characters can see what happens after the impact) automatically meant the impactor would also be similarly illuminated. To simplify things I took its phase angle to be zero—fully illuminated—in the rest of the calculation.
The object approaches from right to left as seen by the pilot and impacts near the Earth's limb. The simplest case is to assume its approach vector is orthogonal to Earth/observer axis, in which case its distance will be roughly the same as the 35,785 km altitude of geosynchronous orbit throughout the approach. After trying several alternatives, I settled on an impactor with a diameter of 5 kilometres and a density about that of a stony asteroid, 3 grams per cubic centimetre. If the object reflects light about as well as a bright asteroid, roughly 40%, then viewed from geosynchronous altitude it will have a visual magnitude of about −5, which, under the circumstances, should be not only easily visible but striking. Flashes from the antennas of Iridium satellites brighter than magnitude −6 can be easily seen in broad daylight through the Earth's atmosphere, so −5 ought to do the job against the black of space.
From the viewing distance and diameter of the object, the apparent diameter is easily calculated (see magnitude.pl for details). Our 5 km object viewed from geosynchronous orbit will subtend asin(5/35785) degrees, or about 29 arc seconds. Human visual acuity is generally in the range of 30 to 60 arc seconds—objects below the limit appear pointlike. But many pilots have exceptional vision, so I left it uncertain.
Having chosen the diameter of the impactor to obtain the required visibility and the kinetic energy required, we can solve for the velocity since the diameter and density trivially determine the mass. This is performed by the program impactor.pl in the download archive, which determines the velocity to be about 3×107 metres per second, or about 10% of the speed of light, precisely what I was aiming for. At that velocity, it takes about 0.4 seconds to traverse the 12756 km diameter of the Earth or, “no more than a second” for two diameters.
This may be overstated. Extreme thermophilic bacteria in the deep hot biosphere might survive and eventually repopulate the oceans (which will condense out after a few thousand years) and surface. But it'll take 'em a while to evolve enough to go looking for pay-back.
At the distance at which Earth orbits the Sun, escape velocity from the solar system is about 42 km/sec (assuming you start outside the Earth's own gravitational field). Any object moving more rapidly is not gravitationally bound to the Sun—it's just “passing through”, and hence cannot be said to belong to the solar system.
Likewise, at the Sun's distance from the centre of the Milky Way galaxy, any object moving faster than about 500 kilometres per second will escape the galaxy and shoot off into intergalactic space. The impactor, as mentioned above, is traveling many times this speed, as Yutaka proceeds to compute.
Although the effects of special relativity can be neglected for the kinds of arm-waving we're engaged in here, most physicists would deem a velocity of 10% of the speed of light “relativistic”, since precision measurements can easily detect the differences from Newtonian kinematics. Here Pete is pondering the probability of a natural object such as an asteroid acquiring the impactor's velocity through a random sequence of gravitational encounters with massive bodies. Gravitational slingshots happen all the time—Jupiter throws asteroids and comets into the Sun or out of the solar system, and occasionally into itself. But, as I've said elsewhere, “Space is called ‘space’ because there's so much space there.” And that's precisely the difficulty. Once a gravitational encounter accelerates an object to greater than the escape velocity of the star system in which it originated, it sails off into interstellar space, which is so empty it beggars the imagination. The probability of a second slingshot encounter over the age of the universe is negligible, and the likelihood of a sequence which could add up to a velocity of a tenth of the speed of light is effectively zero. At the moment the velocity exceeds escape velocity for its galaxy of origin, it would first have to encounter another galaxy before any further acceleration could occur. One can manufacture toy models of coalescing black holes which might accelerate an asteroid to such speeds but, even if you admit that remote possibility, the chance of such an object then managing to hit an infinitesimal target such as the Earth is indistinguishable from zero. Which leaves the other possibility….
This would be dramatic and would easily bring down satellites in low Earth orbit. The total mass of the atmosphere is about 5.136×1018 kg, while the oceans contain approximately 1.37×1021 kg of water. Boiling off the oceans would thus increase the atmospheric mass by more than 250 times, increasing its density at LEO altitudes more than enough to drag the satellites to their doom.
It … is … alive!