Ok lets see if we can get this to match. I get the gamma factor to be

gam = 0.9999999985758079

with this, earth sees any time interval on new horizons t' to be

\detla t' = \delta t / gam

longer than the time t on earth. Integrating both sides above over time, keeping velocity constant, we get

t' = t/ gam = 9.5 yr / 0.9999999985758079 = 9.500000013529824 yr

this gives an difference between our observed "new horizons time" and out local time of

t' - t = t/gam - t = (1/gam - 1) t = 0.4 s

we disagree by about a factor of 45000.

It's been a while since I took a physics course, but:

t' = t * (1-v^2/c^2)^(1/2)

t' = 9.5 * (1-16000^2/299792458^2)^(1/2)

t' = 9.5 * 0.99999999857...

(t-t') years = 426.7 milliseconds

But the transmission time delay is still only 4.5 hours, not 9.5. The onboard clock has simply drifted out of sync with one on Earth.

Whoa! I hadn't thought about that. The speed of New Horizons hasn't been constant, of course, but the accumulated time difference is probably close to what you calculated. There would also be a correction due to it being farther out of the Sun's gravity well, but that's a much trickery calculation.

(Just a note: His math is off.)

You also have to account for the dilation caused by the force of launch itself.

But then you have to subtract the dilation of Earth.

> There would also be a correction due to it being farther out of the Sun's gravity well

Because it is in free-fall relative to the Sun there is no Solar gravitational dilation.