In Lord of the Rings, Shadowfax is described as very fast. Is it possible to deduce his speed from the time taken to travel between, for instance, Meduseld and Minas Tirith?
What's his maximum possible speed? I don't think we're ever given enough information to determine that. But, on one occasion we do we see him trying to get from one place to another as quickly as possible -- when Pippin and Gandalf ride him from Dol Baran (where they camped with the Rohirrim shortly after leaving Isengard) to Minis Tirith.
According to Appendix B ("The Tale of Years"), they leave Dol Baran during the night on March 5, and arrive at Minis Tirith at dawn on the 9th. When they first meet, Gandalf tells Denethor that it was a journey of 150 leagues (517 mi, 833 km). Since they only travel at night, that's 4 nights of riding. So Shadowfax averaged about 129 miles (208 km) per night, or 10.7 miles per hour (17.3 kph) (if they rode for 12 hours per night).
An average, modern, non-Mearas horse can travel about 20-30 miles per day, so that makes Shadowfax about 5 or 6 times faster.
If his top speed is likewise proportional to that of a typical horse, that means Shadowfax can run 150 miles per hour (241 kph) or faster. That seems.. er.. unlikely, so it's probably a combination of speed and endurance.
While this question is old, I find the currently accepted answer to be missing some little details for calculating more accurately a speed (though the general logic has some merit).
Relevant points to making the calculation of Shadowfax's speed are:
- Distance: Gandalf states to Denethor that he rode "from Isengard, one hundred and fifty leagues." But the departure when Shadowfax was moving at great speed occurred from Dol Baran, which according to this map is about 20 miles from Isengard. So 150 leagues x 3 miles to a league = 450 miles - 20 miles already traveled to Dol Baran = 430 miles to Minas Tirith from Dol Baran.
- Time frame: The dates given in "Appendix B: The Tale of Years" of The Return of the King—Gandalf leaves Dol Baran for Minas Tirith on March 5th and arrives on March 9th. That is 5 days inclusive as far as time frame, as he travels on the 5th, 6th, 7th, 8th, and 9th. But he is not travelling all the time...
- Travel periods: In Book IV, Ch. 1: "Minas Tirith," The Return of the King, by Pippin's testimony, Gandalf was only travelling at night: "Another day of hiding and a night of journey had fleeted by," which was the 4th night, for he had previously reckoned up to "the third night since he looked in the Stone" at Dol Baran. On the 3rd night they were travelling through the "land of Anórien" in Gondor, and in the morning "twilight" of the 4th night were passing Rammas Echor, the wall of Pelennor, 10 miles from Minas Tirith, and they "rode to the Great Gate of the Men of Gondor at the rising of the sun," dawn of the 4th night.
Travel leg times: It was 4 nights, but the first night (of March 5th) was not a full night's ride.
For the first night, we know that evening was "deepening" on the 5th when Gandalf tells Merry they will ride "for a few hours" before stopping at Dol Baran, where they made camp "two hours or so before the middle of the night" (Book III, Ch. 11: "The Palantír," The Two Towers). When exactly Pippin's episode happened and the Nazgûl flew over is not clear, but it must have been before midnight since it still had to be the 5th when Gandalf leaves (as Théoden arrived at Orthanc at noon on the 5th and the parley with Saruman had occurred, according to "Appendix B," all the same date as Gandalf leaving Dol Baran). While approaching the Westfold dales, there are still "some hours" till "daybreak," which is when Gandalf was going to stop. Gandalf says it will be two days and they will see "the walls of the tower of Denethor." Now it is the early morning of the 6th, and "Appendix B" states arrival is not till the 9th, so two days seems too soon (but see toward end of post). But not mapped out (M = Midnight; D = Dawn; S = Sunset):
S...M...D.....S...M...D.....S...M...D.....S...M...D 5th | 6th | 7th | 8th | 9th night 1 | |night 2| |night 3| |night 4|
Based on the location of Middle-Earth to our Earth (Minas Tirith being approximately at the same latitude as Florence, Italy), then March nights are approximately (based on 2016 data) between 9.5 hours long and 12.5 hours long, depending upon how deep the twilight was before Gandalf would leave and before he would stop. It seems daybreak was the typical end target (1st night and 4th night for sure), but unclear if sunset was the typical departure time on the long nights.
If so, that would be the 12.5 hours per night, except the first night was just prior to midnight, so about 7.5 hours. So we have 7.5 + (3 x 12.5) = 45 maximum hours of travel time. It might have been less hours. Waiting until a deeper darkness before departing but still using sunrise to stop results in about 11.5 hour night, so 7.5 + (3 x 11.5) = 42 hours.
So with 12.5 hour nights, we have 430 miles / 45 hours = 9.55 mph average.
But with 11.5 hour nights, 430 miles / 42 hours = 10.23 mph average.
A Specific Instance
It seems there were particular points where Shadowfax did bursts of speed, but even when not going full speed, he was often supposedly travelling a higher clip than the averages above show. This observation comes from Chapter 11: "The Palantír," where it also stated it took less than an hour for Shadowfax to go from Dol Baran to the Fords of Isen, which is about 20 miles, yet that was not doing top speed like during the burst at the end of the chapter. If 20 miles was done in less than an hour, then at "trot" speed, he was travelling about 25-30 miles per hour. It just happens that speed matches well the typical gallop pace of a horse and Gandalf's testimony of this leg of the journey that Shadowfax was "running now as fast as the swiftest horse could gallop."
More Calculation (the gallop speed—what the OP asked for)
If the typical horse's gallop pace is 25-30 mph, and the typical trot about 8 mph, then a ratio between those helps establish a possible maximum for Shadowfax.
At 25/8 the gallop is 3.125 times faster than a trot, so Shadowfax would do his gallop at 25 (his trot speed) x 3.125 = about 78 mph. If 30 is his trot pace, then 30 x 3.125 = about 94 mph.
So his maximum speed is probably somewhere between 78 to 94 mph, but still not likely for a long distance...
Back to the Travel Time
In the chapter "Minas Tirith," we also know that Shadowfax at one point in the 4th night journey came, briefly, to a dead stop. This was just after Gandalf had actually urged him to speed up. Both points testify to the fact that:
- In all probability, some (brief) stops occurred during the nights
- In full certainty, Shadowfax was not travelling maximum speed at all times during the nights of travel.
His average of about 10 miles an hour, as one comment with link pointed out, matches an average of some of the best known endurance horses. Only Shadowfax can go days like that and tire little to none. The night journeys likely consisted of periods of stopping, walking (for Shadowfax, probably about 12 mph), trotting (25 mph), and galloping (78 mph).
But even then, to only average 10 mph seems slow, especially given the descriptions of the speed he was travelling when he was travelling. That implies that either he had a number of stops in the night for brief rests, drinks of water, hiding from enemies, etc., and/or Gandalf was not always travelling the whole night, sticking mainly to the darkest hours thereof.
That these "delays" were going on seems self-evident, because at his trot speed only (25 mph) Shadowfax should have covered the distance of 430 miles in just over 17 hours, or within 2 nights time. That would be about the two days Gandalf had originally told Pippin. This gives further confirmation about the general speed of Shadowfax. Gandalf is not one to err in calculations, so likely some things delayed the journey at points, but when Shadowfax was moving, it was often trotting and at times galloping.
On page 202 of The Two Towers I'm reading, chapter 11 "The Palantir", end of book III. Pippen asks Gandalf
"How fast is he going? Fast by the wind, but very smooth. And how light his footfalls are!"
"He is running now as fast as the swiftest horse could gallop, but that is not fast for him."
This seems to imply that Shadowfax is traveling at his trot speed which can be maintained for many miles by normal horses. The fastest horse was clocked by Guiness "having averaged 43.97 miles per hour (70.76 km/h) over a two-furlong (0.25 miles (402 m)) distance in 2008" Horses without riders have been clocked at "close to 55mph" That being said a typical horse trot is maybe around half as fast as his gallop, I don't have any horses so I can't measure. So if Shadowfax trots around 45 or even 55mph he can probably hit over 100mph. Pretty crazy but he IS a magical horse king ridden by a wizard!