# Visibility of a Red Dwarf Companion Star

An interstellar traveller stands upon the surface of an alien planet beneath the light of an unfamiliar sun. The planet upon which he stands orbits one star of a binary system in which one is a G-class yellow dwarf and the other is an M-class red dwarf.

While they are indeed orbiting their common barycentre, the stars are sufficiently distant from one another that the red dwarf companion does not interfere with the orbit of the planet, nor does it contribute sufficient insolation to affect the planet's climate.

My question is this: How close must the companion star be for it to appear noticeably more prominent in the sky compared to background stars?

(My assumption here being that at a certain distance a red dwarf will simply blend in with the stars behind it.)

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A really bright early M-type main-sequence star (something like M0V spectrally, Lacaille 8760 being a good example) might have a visual absolute magnitude of around +9. While I don't know exactly how bright you need this star to be relative to the background, I think a good requirement would be that it is visible during the day, which isn't true of any non-Sun stars visible from Earth and might stand out as unusual. This occurs at an apparent magnitude of about -4, which is also about as bright as Venus.

The equation relating difference in absolute magnitude M and apparent magnitude m to distance d (in parsecs) is:

$$5 + (m-M) = 5*log_{10}(d)$$

so the distance to the red dwarf star we need to achieve this is about 0.025 parsecs, or just over 5,100 AU. (This is well over one hundred times the Sun-Neptune distance, for comparison — as far as I know, this shouldn't cause any stability problems in the system whatsoever.)

If you have a fainter star, you need to move it proportionally closer — quartering the intrinsic visual-band luminosity of the star will drop its distance by a factor of two.

If you want the star to appear brighter, an increase in brightness by one magnitude requires the star to be $$\sqrt[5]{100}$$ times as bright, which mandates a reduction in distance by a factor of $$\sqrt[2.5]{100}$$ or about 6.3.

At some point, one or both of these factors will bring the star too close to the inner system to keep it stable, but for your purposes it can probably be quite a long ways out.

• When you say 'intrinsic luminosity' do you mean the bolometric? Actually, that's probably a silly question given that we're talking about visibility... – Arkenstein XII Apr 2 at 2:13
• Since you're only concerned about visual-band apparent magnitude, it's really just the visual-band absolute magnitude (and thus luminosity) that matters. Cooler red dwarf stars will emit somewhat less visible light in proportion to their bolometric luminosity, which can be approximated by Planck's law. – parasoup Apr 2 at 2:16
• Ok, thanks for the clarification! You mention that quartering luminosity halves distance; is it possible to work out with an equation what the class of a M-dwarf with a quarter of the luminosity of an M0V would be? – Arkenstein XII Apr 2 at 2:24
• I'm afraid I don't really know of a simple approximation; you might be able to work something out with the Stefan-Boltzmann law, knowledge of the star's temperature given its subclass, and/or some functions relating these factors like seen here. There's quite a bit of variance between "similar" stars, too. – parasoup Apr 2 at 2:32
• Just for comparison - Proxima Centauri is a smaller M5.5 class star with absolute magnitude of 15.6. It orbits Alpha Centauri at 8700 AU, and, looking from Alpha system, it would be seen only as a faint magnitude 5 star. – Alexander Apr 2 at 18:40