SR Effects on Interstellar Probe: Can Messages Return in 50 Years?

[dbmorpher]

If a probe is sent to the Tau Ceti system and does not accelerate outside the solar system (keeps a constant velocity when passing the heliosphere) will it not experience relitivistic effects? Could it then send a message back to Earth in 50 years for us?

 

[mfb]

All moving objects experience relativistic effects, but if the velocity is small (compared to the speed of light) those effects are small as well. Every probe can always send signals to us (unless it is REALLY far away (billions of light years)).

 

[ghwellsjr]

If a probe is sent to the Tau Ceti system and does not accelerate outside the solar system (keeps a constant velocity when passing the heliosphere) will it not experience relitivistic effects? Could it then send a message back to Earth in 50 years for us?

You can use the Relativistic Doppler formula to calculate when the probe needs to send that message so that it will arrive on Earth 50 years after launch. Take its speed relative to Earth as a ratio of v/c = β and crank it into the formula:

√((1-β)/(1+β))

Then multiply that factor by 50 and that will tell you when the probe needs to send the message.

Here's a couple examples. First for a speed of 0.6c we calculate:

√((1-0.6)/(1+0.6)) = √(0.4/1.6) = √0.25 = 0.5

So the probe needs to send the message when its clock reads 0.5*50 = 25 years.

In this diagram, Earth is shown as the blue line with dots marking each year (same as the coordinate time since this is the Earth inertial rest frame). The probe is shown in black with yearly dots that are spaced farther apart corresponding to a Time Dilation factor of 1.25. At year 25 the probe sends the green message which travels at the speed of light along the 45-degree angle arriving at Earth time of 50 years.

Lest you think this is obvious (the probe sends the message at one half the total time), let's do another one for 0.8c. In this case, the Doppler Factor is 0.333 so we have to multiply that by 50 to get 16.67 as the time the probe needs to send the message:

If you think about it, the slower the probe goes, the closer to 50 years the probe has to send the message because it will get back to Earth very quickly. The faster it goes, the closer to 0 years it has to send the message because its clock is almost stopped in the Earth frame. In this case, the probe will send the message near the coordinate time of 25 years and the coordinate distance of 25 light-years. So it takes the probe about 25 years of Earth time to get to where it sends the message and it takes another 25 years for the message to get back to earth.

 

[dbmorpher]

Thanks George really informative!
Now just to get the grant...