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Assume this Red Giant has the properties of Aldebaran (α Tauri in constellation Taurus) with an absolute visual magnitude of −0.641M. How far would it have to be from Earth in order to possess an apparent visual magnitude of -13m, (appearing as bright as a full moon)? Would this distance constitute a binary system that would screw up Earth's orbit significantly?

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    $\begingroup$ Advanced Magnitude Calculator $\endgroup$
    – Alexander
    Aug 21 '20 at 19:53
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    $\begingroup$ Downvoters & Closevoters: please remember that real world questions are on-topic. It's true that no one is obligated to explain their vote, but Silvirs is a new user - so buck up and help a new OP learn the ropes. Thanks! $\endgroup$ Aug 21 '20 at 19:56
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    $\begingroup$ @JBH, lack of research is a legitimate reason for downvote, on topic or not. $\endgroup$
    – L.Dutch
    Aug 21 '20 at 20:05
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    $\begingroup$ @L.Dutch-ReinstateMonica I get that, but apparent magnitude isn't the most researchable topic for people who haven't had a college-level astronomy class, which I'm assuming based on the Q, and besides... new user without a sole pointing out any deficiencies. $\endgroup$ Aug 21 '20 at 20:10
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    $\begingroup$ @AlexP If we held people consistently to that concept, half the questions on this site would need downvoting. As I told L.Dutch... new user. On the other hand, your first comment looks like an answer. Why didn't you post it? $\endgroup$ Aug 21 '20 at 23:00
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  • One step of magnitude represents a change in brightness by factor of 2.5. (Actually, it is a factor of $\sqrt[5]{100} = 2.512$, because the definition is that a difference of 5 magnitudes corresponds with a change of brightness by a factor of 100.)

  • Brightness decreases with the square of the distance.

  • Absolute magnitude is defined as the apparent magnitude that the object would have at a distance of 10 parsecs.

  • You want the apparent magnitude to be −13. That is 12.34 steps lower than the absolute magnitude of −0.64, which means that the brightness will be higher by a factor of $2.512^{12.34} = 86{,}346$.

  • The square root of 86,346 is 293.8. Since the absolute magnitude is reckoned at a distance of 10 parsecs, the required change in brightness will happen at a distance of 10 / 293.8 = 0.034 parsecs = 0.1 light years = 7019 astronomical units.

  • A distance of 0.034 parsecs or 0.1 light years is about one fortieth of the distance between the Sun and Proxima Centauri; and about half the distance between Proxima and the double α Centauri, to which it is gravitationally bound.

  • So that yes, if Aldebaran would be so close to the Sun the two stars might, quite likely, be gravitationally bound. On the other hand, the gravitational influence of the nearby very bright Aldebaran on Earth would be very small, much smaller than the gravitational influence of Jupiter. On the third hand, it all depends on how fast the two stars are moving relative to one another; they might as well be just passing by, without being gravitationally bound.

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