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I know that the light coming from Proxima Centauri is not bright enough to make it naked-eye visible from the Earth. Is the Sun naked-eye visible from Proxima Centauri?

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Well, there's two things we'll need for this: apparent magnitude (the brightness that an object appears to have) and absolute magnitude (the actual brightness an object has). Both of these scales are logarithmic, with brighter objects being lower and dimmer objects being higher. Astronomers have determined that the Sun's absolute magnitude is 4.83. Knowing this, we can find the apparent magnitude of the Sun from Proxima Centauri's location. Apparent and absolute magnitudes are related by the equation:

$$M = m - 5 (\log_{10}{d}-1)$$

Where $M$ is the absolute magnitude, $m$ is the apparent magnitude, and $d$ is the distance, in parsecs. Astronomers have determined that Proxima Centauri is 1.3 parsecs from us. So the apparent magnitude can be determined as:

$$m = 4.83 + 5(\log_{10}{1.3}-1) ≈ +0.4$$

As made clear in this paper, most humans can see objects with apparent magnitudes as dim as $5$ without using tools. So yes, it is certainly visible, and would be quite bright. It is between Procyon and Achernar, the 9th and 10th brightest star on Earth's night sky.

For a comparison, Proxima Centauri has an apparent magnitude of $+11.13$ from Earth's perspective. If we want to compare the two, we could use this formula:

$$v_b = 10^{0.4 x} = 10^{0.4×(11.13-0.4)} \approx 19588$$

So the Sun would appear almost 20,000 times brighter from Proxima Centuari than PC appears from the Sun.

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    $\begingroup$ When comparing it to the stars of our night sky, which one would be comparable in brightness then? $\endgroup$ – polemon Aug 25 '16 at 8:39
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    $\begingroup$ @polemon: Procyon, at +0.38 $\endgroup$ – MSalters Aug 25 '16 at 9:24
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    $\begingroup$ TIL"The brighter an object appears, the lower its magnitude value" This answer did not make sense to me until I learned this. $\endgroup$ – Régis B. Aug 25 '16 at 12:20
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    $\begingroup$ It might be worth in this case to give a little crash-course on apparent magnitude: 0.0 is the magnitude of Vega, the brightest star visible in the night sky, 7.0 the dimmest still visible with the naked eye (in a perfectly dark and clear night, far away from natural light sources) $\endgroup$ – Philipp Aug 25 '16 at 13:49
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    $\begingroup$ @Philipp: Nitpick: Vega is the fifth brightest star in the night sky, after Sirius, Canopus, $\alpha$ Centauri, and Arcturus. These four stars actually have negative apparent magnitudes (Sirius's magnitude is -1.5 or so.) $\endgroup$ – Michael Seifert Aug 25 '16 at 13:53
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Alpha Centauri A and B happen to be rather similar to Sol, and their absolute magnitudes are 4.38 and 5.71 respectively (Wikipedia). Add them together and you get absolute magnitude 4.10 (the scale is logarithmic, and backward). Sol, with absolute magnitude 4.83, should look 0.73 magnitude dimmer than αCen at the same distance, so magnitude +0.46, quite bright.

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  • $\begingroup$ Your answer isn't exactly right. The actual brightness would be 1.4 times brighter than what you claim. $\endgroup$ – Sir Cumference Aug 25 '16 at 21:21
  • $\begingroup$ Oh? What did I miss? Did I use the wrong base (10^-0.4) for the logarithms? $\endgroup$ – Anton Sherwood Aug 26 '16 at 7:42
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    $\begingroup$ The difference between magnitude 0.4 (your answer) and magnitude 0.46 (my answer) is accounted for by the distance between Proxima and Alpha A/B, which I ignored for simplicity. It's not a factor of 1.4. $\endgroup$ – Anton Sherwood Aug 27 '16 at 3:56
  • $\begingroup$ Ah, sorry, I misread it as the apparent magnitude being 0.73. $\endgroup$ – Sir Cumference Aug 27 '16 at 4:02
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The key to this is the so called Absolute Magnitude, which represents the visual magnitude from a distance of 10 parsecs (about 32 light years). The sun is much brighter than Proxima Centauri. It has an absolute magnitude of 4.8, and at a distance of 4 light years (the distance of Proxima), it would be be somwhat brighter than 1st mag, and so very easily visible with the unaided eye.

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  • $\begingroup$ @Patrick .. This is a website you might enjoy: bdm.id.au/localspace/systems.html . Click on any star of your choosing and you will see a pretty good rendition of what the night sky would look like when looking back toward the sun. $\endgroup$ – Jack R. Woods Aug 29 '16 at 17:32

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