First order estimate: The Horse
The Roman Empire at its peak (117 A.D.) is estimated to have encompassed 5 million square kilometers. Now, the Mediterranean sea was in the middle of it, so the speed of an early ship is also involved, but let's be generous and stick with the horse. The average top speed of a horse (because there are many breeds and they don't all do the same thing) is about 88 kph.
OK, so our goal is to visit the center of each and every one of those square kilometers on the horse at top speed. No, you obviously can't do that in reality as some kilometers would need to be passed over again to get to others, but bear with me, it's a simplistic way of analyzing the scope of this baby. Please keep your hands and feet inside the ride at all times.
5,000,000 km2 / 88 kph = 56,818 hours to visit each square kilometer or about 6.5 years.1
Let's test our theory
The U.S. had the Pony Express, but it really didn't hit its stride until the Railroad! The area of the continental U.S. is 8,080,464.3 km2 and the average top speed of the rail in 1860 was 80.5 kph.
8,080,464.3 / 80.5 = 100,378.4!
OK! Our solid two points of experimental overly-simplified data suggests that the railroad would be slower than the horse — except — that assuming the horse could actually hold out at 88 kph all that time is really stretching reality!
So I humbly declare that these numbers are on the right track (ah-ha).
One more check: From the Earth to the Moon
So, the volume of the Earth is about 1 trillion cubic km and the volume of space described by the orbit of the moon in three dimensions is V=4𝛑r3/3 where r=385,000 km so V=477,874,465,321,626,227 cubic km (yeah, 4.8 quadrillion...) and we remove the Earth's volume from that to get 477,873.5 trillion cubic km and Apollo 10 was booking through space at a top speed of 39,897 kph... and we want to visit each and every one of those cubic kilometers!
477,973,465,321,626,227 / 39,897 = 11,977,679,156,869
Yup, that's 11 trillion hours of watching Gilligan's Island! And we've also lost our working theory with the horse that can't actually keep up that long. What we have proven is an axiom from Douglas Adams: "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."
But, to be fair, there's a whole lot less to see in space. I mean, what's the point of visiting all those cubic kilometers?2 Let's assume that railroad number is absolutely spot-on,3 which means we really don't care about 99.999999% of the volume of space.4
But what speed are we talking about on a galactic scale?
OK, so our magic number is 100,378 and we're ignoring 99.999999% of the volume of space and the estimated volume of the Milky Way is 6.65X1051 cubic kilometers. So, the speed you need is:
.00000001 * 6.65x1051 / 100,378 = 6.625x1038 = 614x1027 times the speed of light
or approximately warp factor 8,499,379,494. (AKA warp factor 8.5 billion.)
Which absolutely proves Mr. Adams' axiom and demonstrates that trying to match reality with your fiction is frequently a waste of time.
Why on earth should I trust this number?
Because the Romans didn't use horses to visit every square kilometer of their empire. What I did was use an arbitrary ratio5 that compared two very real and very practical numbers in such a way that I could try to normalize the effects over time.
If anything, what this proves is that humanity must move at exponentially faster speeds to manage growing empires.
The surprise in all this to me was that the railroad actually slowed down compared to the horse. But what I'm not comparing is the ability of the railroad to carry vastly more goods than the horse could. So while my magic number increased, a heretofore unknown ratio describing the increase in logistics vastly improved.
My conclusion? You're looking at the wrong number. Speed isn't that relevant. Logistics is. You should be looking at the ability to haul a boatload of freight between solar systems. Because in the end, far flung islands of the empire will rule themselves more according to how well the government was set up than how fast communication (and
assassins investigators) can travel.
1 *And to make that work you had to urinate from the back of the infinitely enduring horse while eating Ofellas thrown at you by vapid admirers. but hey, it's just a first-order estimate.*
2 Other than bragging rights. NEVER underestimate the value of high-quality bragging rights.
3 Stop laughing, my number is just as good as yours! My momma can kick yo momma's butt, too! And yours wear combat boots! Yeah! Yo momma so ugly the TIDE won't take her out! What were we talking about...? Oh, yeah, angels weeping over the ridiculous number. Yo momma still ugly....
4 And I dare anybody to prove that number wrong! Hah!
5 A remarkably arbitrary ratio.