How Fast Would a Ship Be Going at the Halfway Point to Mars at Continuous Acceleration
The basic principle behind every high-thrust interplanetary space probe is to accelerate briefly and then coast, following an elliptical, parabolic, or mildly hyperbolic solar trajectory to your destination, using gravity assists whenever possible. But this is very slow.
Imagine, for a moment, that we have a spacecraft that is capable of a constant 1g ("one gee" = 9.8 m/s2) acceleration. Your spacecraft accelerates for the first half of the journey, and then decelerates for the second half of the journey to allow an extended visit at your destination. A constant 1g acceleration would afford human occupants the comfort of an earthlike gravitational environment where you would not be weightless except during very brief periods during the mission. Granted such a rocket ship would require a tremendous source of power, far beyond what today's chemical rockets can deliver, but the day will come—perhaps even in our lifetimes—when probes and people will routinely travel the solar system in just a few days. Journeys to the stars, however, will be much more difficult.
The key to tomorrow's space propulsion systems will be fusion and, later, matter-antimatter annihilation. The fusion of hydrogen into helium provides energy E = 0.008 mc2. This may not seem like much energy, but when today's technological hurdles are overcome, fusion reactors will produce far more energy in a manner far safer than today's fission reactors. Matter-antimatter annihilation, on the other hand, completely converts mass into energy in the amount given by Einstein's famous equation E = mc2. You cannot get any more energy than this out of any conceivable on-board power or propulsion system. Of course, no system is perfect, so there will be some losses that will reduce the efficiency of even the best fusion or matter-antimatter propulsion system by a few percent.
How long would it take to travel from Earth to the Moon or any of the planets in our solar system under constant 1g acceleration for the first half of the journey and constant 1g deceleration during the second half of the journey? Using the equations below, you can calculate this easily.
Keep in mind that under a constant 1g acceleration, your velocity quickly becomes so great that you can assume a straight-line trajectory from point a to point b anywhere in our solar system.
Maximum velocity is reached at the halfway point (when you stop accelerating and begin decelerating) and is given by
The energy per unit mass needed for the trip (one way) is then given by
How much fuel will you need for the journey?
hydrogen fusion into helium gives: Efusion = 0.008 mfuel c2
matter-antimatter annihilation gives: Eanti = mfuel c2
This assumes 100% of the fuel goes into propelling the spacecraft, but of course there will be energy losses and operational energy requirements which will require a greater amount of fuel than this. Moreover, we are here calculating the amount of fuel you'll need for each kg of payload. We would need to use calculus to determine how much additional energy will be needed to accelerate the ever changing amount of fuel as well. The journey may well be analogous to the traveler not being able to carry enough water to survive crossing the desert on foot.
Now, let's use the equations above for a journey to the nearest stars. There are currently 58 known stars within 15 light years. The nearest is the triple star system Alpha Centauri A & B and Proxima Centauri (4.3 ly), and the farthest is LHS 292 (14.9 ly).
I predict that interstellar travel will remain impractical until we figure out a way to harness the vacuum energy of spacetime itself. If we could extract energy from the medium through which we travel, we wouldn't need to carry fuel onboard the spacecraft.
We already do something analogous to this when we perform a gravity assist maneuver. As the illustration below shows, the spacecraft "borrows" energy by infinitesimally slowing down the much more massive Jupiter in its orbit around the Sun and transferring that energy to the tiny spacecraft so that it speeds up and changes direction. When the spacecraft leaves the gravitational sphere of influence of Jupiter, it is traveling just as fast as it did when it entered it, but now the spacecraft is farther from the Sun and moving faster than it would have otherwise.
Of course, our spacecraft will be "in the middle of nowhere" traveling through interstellar space, but what if space itself has energy we can borrow?
Source: https://cosmicreflections.skythisweek.info/2019/09/04/space-travel-under-constant-1g-acceleration/
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