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The transmitting speed of the New Horizon probe is $4$ kb/s. Thus, NASA states that it will take around 16 months to download all of the data that the probe gathered in its flyby. Why is this?

P.S. Should I ask this question in this forum or in the Engineer forum? Thank!

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The bit rate of a channel depends on the signal to noise ratio (and their statistics but I will assume these are fixed). As the received signal for a given system declines with range the bit rate of the channel falls also (conditioned on a fixed noise power). The Shannon-Hartley theorem quantifies this for a channel subject to Additive Gaussian White Noise: $$ C=B\log_2\left(1+\frac{S}{N} \right) $$ where $C$ is the bit rate (bits/s), $B$ the bandwith of the channel in Hz, and $S/N$ is the signal to noise power ratio.

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