Seven cosmic messengers

By on 10/10/2020

Let us suppose we travel from Earth to the furthest observable point in the universe. We have seven satellites on our spacecraft, used to keep communications between us and the Earth. Let’s suppose that the speed of the satellites coincides with that of light, or in any case equal to a speed whose difference with c is negligible, while the speed of the spacecraft is v = 2 / 3c. The satellite, once it reaches Earth orbit, transmits the information we have loaded into its memory, then heads back to us to collect the new information. Meanwhile, within 24 hours of each other, we launch all the satellites.
The time each probe takes will be given by the formula

t = \frac{y_1+y_0}{c}

where y_0 is the distance traveled on the outward journey (or if you prefer the relative position of the spacecraft respect to the Earth at the time the first probe was launched), y_1 the distance of the return (or the position of the spacecraft when the first probe returns) and c is the speed of the probe.
In the meantime, the spacecraft has also moved by the y - x segment and the time taken by the spacecraft is given by

t = \frac{y_1-y_0}{v}

These two times, however, are the same, so it is easy to derive the relationship between the initial and final position of the spacecraft during the travel period of the first probe:

y_1 = \frac{c+v}{c-v} y_0

So the path that the first probe will travel on its second journey will be

y_2 = \frac{c+v}{c-v} y_1 = \left ( \frac{c+v}{c-v} \right )^2 y_0

and on the umpteenth journey will be

y_n = \left ( \frac{c+v}{c-v} \right )^n y_0

Suppose, then, that the first probe is sent after two days, that is

y_0 = v \cdot t_{2 \, days}

The second probe will be launched after 3 days, the third after 4 and so on. So in general the initial starting point for probe k will be

y_0^{(k)} = v \cdot t_{1+k \, days}

At this point, setting c = 1 and the time in days, we can obtain a table like the one in the image below:

Now let’s try to ask ourselves this question: how many times do I have to launch a probe before reaching, for example, Proxima Centauri b, the closest exoplanet, at a distance of about 4 light years, or 1460 days?
If we look at the table, we conclude that, before reaching Proxima Centauri b we will launch the first and second probe 4 times, while all the others 3 times.
If instead we don’t want to use the table, but to apply a formula, we will have to invert the formula of y_n using logarithms:

\log n = \log \frac{y_n}{y_0^{(k)}} / \frac{c+v}{c-v}

About Gianluigi Filippelli

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