# how fast does time pass for voyager 1 and 2, relative to Earth?

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A second on Voyager 1 or Voyager 2 is still registered as a second.  But I suppose you want to know the time dilation on there.

Voyager 1 is traveling at around 17 km per second, which puts it at

17/(3 * 10^5)  c =>

5.66666.... * 10^(-5) c

sqrt(1 - (v/c)^2) =>

sqrt(1 - (5.66666 * 10^(-5) * c / c)^2) =>

sqrt(1 - 5.66666....^2 * 10^(-10)) =>

sqrt((10^10 - (17/3)^2) / 10^10) =>

sqrt(10^10 - 289/9) / 10^5 =>

10^(-5) * sqrt(9 * 10^10 - 289) / 3 =>

(1/3) * 10^(-5) * sqrt(90,000,000,000 - 289) =>

(1/3) * 10^(-5) * sqrt(89,999,999,711) =>

0.9999999983944444431555401213873.....

From our perspective, every second that passes on Earth, 0.99999.... seconds pass on Voyager 1.  To put this in perspective, if we took 2 identical and synchronized clocks, placed one on Voyager 1 and kept the other on Earth, after 622,837,370 seconds on Earth, the clock on Voyager 1 would have counted 622,837,369 seconds.

That works out to 1 second of "loss" for every 19 years 269 days 0 hours 22 minutes 50 seconds.  That's assuming a year of 365.25 days, 24 hours per day, 60 minutes per hour, 60 seconds per minute.

EDIT:

Of course, I rounded.  So let's see what happens if I don't do that

c = 299792458 m/s

Voyager is traveling at closer to 16900 m/s

16900 / 299792458 = n

sqrt(1 - n^2) =>

sqrt(1 - (16900/299792458)^2) =>

0.9999999984110800861902969516832...

629,358,340.4102045722717942150627.... seconds on Earth before there is a 1 second difference between the hypothetical synchronized clocks.

19 years 344 days 11 hours 45 minutes 40.4 seconds.

Voyager 2 is traveling at around 15.2 km/s

sqrt(1 - (15200/299792458)^2)

That's a second difference every 778,008,291.3428130536952871620136... seconds on Earth

24 years 238 days 17 hours 24 minutes 51.3 seconds.

• Time is never absolute....

• Time passes the same for them as for us. Their speed is so slow compared to the speed of light that relativity effects are not noticeable.

• There are two effects for time dilation: gravity and speed. The Voyagers are far from the Sun and Earth, so they are in a very weak gravity field.  That means that the rate of time is faster for them. The second effect is from speed.  Since the Voyagers are going very fast, the rate of time is slower for them.

The gravity effect is about +0.489 seconds per year.

The Voyager I speed is about  17 km/s, for a time dilation of -0.051 seconds per year.

So the gravity effect wins out.  The overall effect is that the rate of time on Voyager I is faster than Earth, by +0.438 seconds per year. For Voyager II, it's +0.428 seconds per year.

A radio signal from Voyager, operating at a perfect 100 GHz, would seem to us to be at 100,000,001,704.82 Hz (after correcting for Doppler).