Time dilation on Jupiter's surface due to rotation

TIME DILATION ON JUPITER’S SURFACE DUE TO ROTATION

INTRODUCTION

Jupiter is the 5^th planet in our Solar System. It completes one rotation about its axis in 9 hours 55 minutes and 30 second. It is also the fastest rotating planet in our Solar System. Consider the planet Jupiter which rotates at a certain velocity and another analogous Jupiter which doesn’t rotate. According to the Special Theory of Relativity, a clock on rotating Jupiter would run slower than that on a non-rotating Jupiter. We’ll find the time gained by rotating Jupiter relative to the stationary one.

ASSUMPTIONS

Jupiter is not revolving around the Sun.

Jupiter is a perfect homogeneous sphere.

The effect of Gravitational time dilation is negligible.

CALCULATION

The Angular velocity is given by,

ω = 2π/T [rad/s]

T – Rotational time period [T = 9 hours 55 minutes 30 second = 35730 s]

ω = 2*3.14/35730

ω = 1.7585*10^-4 rad/s

The angular velocity is same at all points on Jupiter since it doesn’t exhibit differential rotation. But the tangential velocity on surface varies with the distance from the center.

The tangential velocity is given by,

v = R*ω [m/s]

R – Average radius of Jupiter [R = 69911 km]

v = 69911*1000*1.7585*10^-4

v = 12,293.8493 m/s

According to the Special Theory of Relativity, the time dilation equation is,

t' = t/γ [s]

t’ – Actual time or Moving observer’s time. [s]

t - Proper time or Stationary observer’s time. [s]

γ – Relativistic gamma factor, γ = 1/√ [1-(v/c) ²]

c - Velocity of light [c = 3*10⁸ m/s]

t' = t*√ [1-(v/c) ²]

t' = t*√ [1-1.6793*10^-9]

t' = t*√ [0.9999999983207]

t' = t* 0.99999999916035

CONCLUSION

We can observe that proper and actual time isn’t the same which proves that time dilates on rotating Jupiter relative to the non-rotating one. We’ll consider 5 different t’ values and calculate t value. The larger the t’ the more is the difference between t and t’. Thus rotating Jupiter will gain time over the non-rotating one. In fact it’ll gain 26 milliseconds in one year.

Time	t’ [Non-rotating Jupiter] (s)	t [Rotating Jupiter] (s)	Difference (s)
1 minute	60	59.999999949621	0.000000050379
1 hour	3600	3599.99999697726	0.00000302274
1 day	86400	86399.9999274542	0.0000725458
1 month	2592000	2591999.99782363	0.00217637
1 year	31536000	31535999.9735208	0.0264792