I'd like to specifically single this point out. Ever since the first Okuda Star Trek Chronology book, it was assumed that the Valiant had warp drive, and that it was launched two centuries before the Enterprise found its recorder buoy (i.e. it was launched and then lost at the edge of the galaxy in the same year.) Even the model Greg Jein built for it had warp nacelles, to go along with the model he also built for Cochrane's warp ship, because he also assumed it must have had warp drive. But there's no evidence at all that the Valiant had warp drive, or how long it was already in space before it reached the edge of the galaxy by whatever means brought it there. Remember that this was just the second pilot of the show; they had no idea how far into the future Star Trek took place at the time. Saying that the ship was lost 200 years before would be meaningless if it was 200 years before the 28th century. But besides that, if Earth already had advanced sleeper ships in 1996 that could be lost in interstellar space only two hundred years before, then a ship with no warp drive could absolutely have reached the edge of the galaxy by 2065 if it had been launched at the same time as the Botany Bay....
Assuming that a sleeper ship from Earth could reach the edge of the galaxy in a mere 69 years from 1996 reveals a certain lack of basic astronomical knowledge as well as forgetting some Star Trek lore (and I don't mean Data's brother).
Space is big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist, but that's just peanuts to space.
https://en.wikiquote.org/wiki/Space
There is a good reason why the adjective for large numbers is "astronomical".
An Astronomical Unit or AU is based on the semi-major axis of Earth's orbit around the Sun but is now defined as exactly:
149597870700 metres (exactly)
https://en.wikipedia.org/wiki/Astronomical_unit#Development_of_unit_definition
That equals 149,597,870.700 kilometers or 92,955, 807 miles.
The planets in the Solar System have somewhat eliptical orbits and so their distances from the Sun vary a bit, bu their orbital semi-major axis in AU are:
Mercury 0.39
Venus 0.72
Earth 1.00
Mars 1.52
Jupiter 5.20
Saturn 9.54
Uranus 19.22
Neptune 30.06
So basically the longest interplanetary voyage in our solar system would be approximately 49.28 AU between Uranus and Neptune when they were on opposite sides of the Sun.
Of course the dwarf planet Pluto was formerly classified as a planet and ranges from 29.658 AU to 49.365 AU from the Sun. Counting Pluto as a planet, an interplanetary voyage between Neptune and Pluto on opposite sides of the Sun could be as long as about 79.425 AU.
The hypothetical Planet X or Planet Nine, if real, would have an orbital semi-major axis of about 400 to 800 AU.
https://en.wikipedia.org/wiki/Planet_Nine
So if Planet Nine turns out to be real, it might be possible for an interplanetary voyage between Neptune and Planet Nine on opposite sides of the Sun to be a slong as about 830 AU.
In TOS "The Changeling" a map of Earth's solar system is seen:
SPOCK: Chart 14A, sir?
KIRK: 14A. (a diagram of our solar system comes up on a screen) Nomad, can you scan that?
NOMAD: Yes.
KIRK: This is our point of origin, the star we know as Sol.
NOMAD: You are from the third planet?
KIRK: Yes.
NOMAD: A planet with one large natural satellite?
KIRK: Yes.
NOMAD: The planet is called Earth?
KIRK: Yes.
That chart shows 8 planets in orbits which seem to share a plane, and a more distant planet in a tilted orbit, presumably Pluto.
So presumably when "Space Seed" was written the longest possible interplanetary voyage would have been imagined to have been about 80 AU between Neptune and Pluto on opposite sides of the Sun.
Aboard the Botany Bay in "Space Seed":
(Men and women are lying in clear-sided compartments, seemingly asleep.) KIRK: Scotty?
SCOTT: Definitely Earth-type mechanism, sir. Twentieth century vessel. Old type atomic power. Bulky, solid. I think they used to call them transistor units. I'd love to tear this baby apart.
MARLA: Captain, it's a sleeper ship.
KIRK: Suspended animation.
MARLA: I've seen old photographs of this. Necessary because of the time involved in space travel until about the year 2018. It took years just to travel from one planet to another.
It is possible that Marla meant that even a "short" interplanetary voyage to Mars or Venus took at least one Earth year and required suspended animation. But assuming that actually only the very longest interplanetary voyages, such as from Neptune to Pluto on opposite sides of the Sun, took as long as one Earth year, to get the speeds of space ships as fast as possible, then we can calculate the speeds involved.
Assuming that a voyage of 80 AU took exactly one Earth year, the average speed in the voyage would be 80 AU per year, or about 0.219 AU per day, or 0.0091261 AU per hour.
Assume that the ship accelerated for half a year, and then turned over and decelerated for half a year.. In that case the maximum speed that the ship reached would be 160 AU per year, or about 0.4380 AU per day, or about 0.0182 AU per hour. And since the ship was able to decelerate from that speed, it would have enough fuel, and reaction mass if necessary, for its atomic engines to accelerate it to twice that maximum speed, and thus to a speed of 320 AU per year, or 0.8461 AU per day, or 0.0365 AU per hour.
When the space ship reached its destination it might gather fuel and/or reaction mass at the destination planet for its return voyage. But if the voyage planners didn't anticipate that fuel and/or reaction mass would be available at the destination planet, the ship would have to carry all the fuel and/or reaction mass necessary for the return voyage.
So if the augments took over a ship that had enough fuel for such a voyage they could have accelerated it to a speed of 320 AU per year using half of the fuel and/or reaction mass, planning to use the other half of the fuel and/or reaction mass to decelerate when they reached the destination star system.
But maybe the augments took along a device like a solar sail or a magnetic sail and planned to use it to decelerate when they reached their destination star. In that case they could have used all of the fuel and/or reaction mass to accelerate to a speed of 640 AU per year, or 1.7522 AU per day, or 0.0730 AU per hour.
And surely such a vast speed compared to present day space craft would be fast enough to reach another star system in two or three centuries. Right?
Wrong. Distances to stars are usually measured in two units, parsecs and light years.
A parsec is defined as the distance at which one AU would have a parallax of one arc second, or a distance of 206,264.806274096 AU. That equals 3.261563777 light years.
https://en.wikipedia.org/wiki/Parsec
A light year is defined as the distance traveled by light and other forms of electromagnetic radiation during one Julian calendar year of 365.25 days. That is exactly 9,460,730,472,580,800 meters or 9,460,730,472,580.8 kilometers or 5,878,625,000,000 miles or 63,241.077 AU or 0.306601 parsecs.
https://en.wikipedia.org/wiki/Light-year
In our region of the galaxy, the typical separation between one star and its nearest neighbor will be several light years. For example, Proximal Centauri, or Alpha Centauri C, is about 4.244 light years or 1.3012 parsecs from Earth and the Solar System.
About 4.244 light years is about 268,395,13 AU and about 1.3012 parsecs is about 268,391.7658 AU. So at a speed of 640 AU per year, it would take a spaceship 419.3621 to 419.36739 years to reach Proxima Centauri.
Our galaxy is a sphere with a diameter over 100,000 light years. Most of the stars are concentrated in the galactic disc which is about 100,000 light years in diameter, and has a thickness of about 1,000 to 2,000 light years near the Sun.
So if the force field at the edge of the galaxy is wrapped around the galactic disc, the shortest distance to it would be to go "up" or "down" away from the galactic plane for a distance of about 500 to 1,000 light years from Earth.
A distance of 500 light years would be 31,620,538.5 AU, and a ship travelling at a speed of 640 AU per year would take about 49,407.09141 years to travel 500 light years. That is 716.04479 times the 69 years that you stated were enough for the Valiant to reach the edge of the galaxy.
If a ship could travel 500 light years, the absolute closest possible distance to the edge of the galaxy, in only 69 years it would be travelling at a speed of 7.2463 times the speed of light. At such speeds voyages to the very nearest star systems would take less than one year. Such a speed would equal 458,268.6739 AU per year, or 1254.67125 AU per day, or 52.27796873 AU per hour. At a such a speed a voyage from Neptune to Pluto on opposite sides of the Sun that was 80 AU in distance would take only 1.530281339 hours, not years. At such a speed, even a voyage from Neptune to Planet Nine 830 AU apart on opposite sides of the Sun would take only 15.87666889 hours, less than one day, and not years.
However, it is possible that in the era of the Valiant ships could reach their destinations in less time that it would take travelling at the speed of light, without travelling faster than light or using warp drive. Possibly they found some type of space warp that a ship could enter and instantly appear in a distant star system, and used a network of those space warps to jump from star to star.
Thus interstellar travel in relatively short time spans could have been possible before the warp drive was invented.
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