### Is the Sun visible from Proxima Centauri to human eyes?

• 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?

• 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.

When comparing it to the stars of our night sky, which one would be comparable in brightness then?

would be another question. @polemon why don't you ask?

@polemon: Procyon, at +0.38

@RobJeffries in many documentation you can read +0.4

TIL"The brighter an object appears, the *lower* its magnitude value" This answer did not make sense to me until I learned this.

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)

@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.)

@Philipp The dimmest possible is disputed. Some sources list it at almost 8.0, while others say about 5.0.