How close from Earth could a wandering flare red dwarf star be without causing havoc?

Rewriting the question to clarify:
The time and location is now in our current Earth. A wandering red dwarf flare star (like Wolf 359) is moving towards our Solar System.
The question: How close could it be to Earth without disrupting our civilization?.

• Reasonably full answer is here: en.wikipedia.org/wiki/Habitability_of_red_dwarf_systems – Mołot Nov 26 '18 at 13:59
• Hi Carlos! I think your question is unclear. From the title and reading through the question, I was thinking your query was about habitable zones and orbital radii. But then, you turn 180deg and toss in the fact that the red dwarf in question is a wandering star! That's a pretty key piece of information! I'm going to VTC until you can edit your title and question body to reflect which of your questions you actually want to ask! – elemtilas Nov 26 '18 at 15:59
• @elemtilas. I have rewritten the question to clarify. – Carlos Zamora Nov 26 '18 at 16:43
• That would depend a lot on trajectory of said red dwarf. I for sure do not want it any closer than 2 light years. – Artemijs Danilovs Nov 26 '18 at 19:58

Considering luminosity

The absolute magnitude of Wolf-359 is 16.65. That is not very bright.

Case 1: The dwarf adds to the planet no more light energy than 1% of the main sun

Lets say we define 1% of the sun as a mostly minimal source of extra heat and energy. If we added that much extra solar energy to Earth, global temps would rise, but probably stabilize near to where they are now due to higher blackbody emissions.

The apparent magnitude of the sun is -26.74. The brightness factor between to magnitude values (where $$\Delta m$$ is the difference in magnitude) is $$10^{0.4\Delta m}$$. Set that equal to a factor of 100 (for 1% of the sun's apparent magnitude), and we get $$\Delta m = 5$$. Therefore, we need the flare star to have apparent magnitude of -21.74.

The formula for apparent magnitude ($$m$$) from absolute magnitude ($$M$$) based on distance in parsecs ($$d$$) is $$M = m - 5\left(\log_{10}d-1\right)$$. This solves for 0.043 AU. This is actually very close, about 20 times farther than the moon, but closer than any other planet.

This suggests that the planet could actually orbit the flare red dwarf, as long as the flare red dwarf orbited the sun in the habitable zone.

Case 2: The dwarf adds to the planet no more light energy than the moon

Let us say we want no appreciable effect on the planet's surface from the red dwarf's radiation. In that case, let us ensure the dwarf is never any brighter than a full moon.

We re-do the calculation, except we want the dwarf's apparent magnitude to be equal to that of a moon, $$m = -12.74$$.

Plug in the rest of the values for $$M = m - 5\left(\log_{10}d-1\right)$$ and we solve for 2.73 AU. This is farther than Mars, but closer than Jupiter.

So if we replaced Jupiter with Wolf 359, Wolf 359 would not be any brighter than a full moon.

Considering the flares

Energy from the flares

From Cwiok, et al, 2006, we see that the optical apparent luminosity of Wolf-359 increases to about magnitude 9 (from its normal 13.54). Peak luminosity is only a few seconds, and within about five minutes luminosity has returned to normal.

These flares would be visually noticable from Earth, but would not provide appreciable heating to disrupt the environment.

X-rays and gammas

Wolf-359 does release X-ray and gamma energy. Both of these are strongly blocked by the Earth's atmosphere. I cannot find a source to determine the magnitude of the X-ray and gamma energy released, so see if either would be a radiation hazard on Earth. However, given that the Earth's atmosphere absorbs both, I strongly doubt Wolf 359 could put out enough energy to cause a problem.

I will keep looking into this, though.

• Hmm, if it seems probable it could be safe, radiation wise, within the solar system the limiting factor would probably be gravity? It is something like two orders of magnitude more massive than Jupiter? – Ville Niemi Nov 26 '18 at 6:42
• @VilleNiemi I remember reading an XKCD "What if" about a black hole wandering acrros the solar system. If I recall correctly, it won't disturb anything much unless it was in a collision course, or nearly about it. – Rekesoft Nov 26 '18 at 9:08
• @Rekesoft I don't know about XKCD, but it is a generally accepted theory that (much more distant) wandering stars disrupt objects in Oort cloud, sending them to inner solar system, sometimes on a collision course with planets, including Earth. – Alexander Nov 26 '18 at 18:12
• if im not wrong, even in case1 earth'd have 0.29 m/s^2 acceleration due to gravity... if you omit that acceleration gets stronger when approaching, youd reach 10% of light speed in only three years and a bit. plus, it may cause the water on earth to move (if im correct). thus i fully agree with Ville Niemi. – KGM Nov 26 '18 at 19:28
• @Rekesoft The perturbances accumulate over time. So the effects of something wandering across the system in some years and something that is part of the system with a stable orbit for billions of years are not actually the same problem at all. – Ville Niemi Nov 27 '18 at 6:12