Factoid Friday – Time Flies

This morning, I awoke to a tweet from astronaut and commander of the International Space Station, Chris Hadfield, saying:

Good Morning, Earth! How did it get to be Friday so soon? Time sure flies at 5 miles per second! (link)

And while I’m sure time does fly while orbiting the Earth 15.7 times a day, there is actually a physical explanation, albeit small, that causes the time on the ISS to slow down: time dilation. So, in my semi-concious stupor, I set about calculating the actual difference in time that Chris Hadfield experiences for his trip on the ISS, and tweeted him back:

@Cmdr_Hadfield At 5 miles per second, your 147 day flight is about 4.5 milliseconds shorter than the time that passes on Earth. (link)

Work

1. The ISS travels at about 5 miles per second (as per Chris’ tweet, the actual value averages 4.7886 miles per second). (cite)

2. Time is dilated when travelling by the gamma factor:

$t = \frac{t_0}{\sqrt{1-v^2/c^2}}$.

Here, $t$ represents the time passes on Earth, for $t_0$, the corresponding time that passes on the ISS. So the quantity we are interested in is $t - t_0$.(cite)

3. Chris Hadfield’s mission on Expedition 35 will last 147 days. (cite)

4. So, plugging in 5 miles per second for ‘v’ (appropriately changed to the same units as ‘c’), and using 147 days for ‘$t_0$‘ (appropriately changed to seconds), we get:

$t$ = 1.27008000045751×10^7 seconds, or 147 days 0.0045751 seconds. (WolframAlpha)

5. Subtract 147 days, and you’re left with 0.0045751 seconds, or about 4.5 milliseconds. To interpret this as 4.5 milliseconds shorter, we need to remember that the resolution of the Twin Paradox applies the time dilation to the twin that was accelerated, in this case, Chris Hadfield.

This post will invariably attract controversy, so in the interest of staunching the flow of criticisms, please refer to this worked example of time dilation.

Disclaimer

Factoid Fridays are tweet sized factoids with appropriate, but sometimes not peer reviewed, citations. For simplicity, I’ve left out unit conversions (i.e. miles per second to meters per second), so those calculations can be considered an exercise to the reader. If you think there is an error in an assumption or calculation, please contact me.