I'm trying to determine what a star of comparable mass and luminosity to the Sun would look like from the surface of the Earth if it were 1 light year away - I chose Alpha Centauri since it is close to us, and similar to the Sun. Could a star like the Sun be that close without exerting significant influence over it, or would they have an orbital relationship? I'm trying to design a star system for a science fiction novel, and would like there to be a star close enough and bright enough to be visible during the day, and cast a shadow at night, but far enough away to provide no heat energy, and be free from the Sun's gravity.
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3$\begingroup$ The title of your questions does not seem to correspond to the body of your question. The visual brightness is straightforward. Alpha Cen is at about 4 light years. At 1 light year it would be 16 times brighter than it appears now. What does "provide no heat energy" mean? Light does transfer heat energy. $\endgroup$– ProfRobSep 15, 2017 at 16:11
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1$\begingroup$ So you'd like a star to be like the full moon in brightness but still quite distant? Maybe better for world building. There would be some heat, but at a full-moon brightness it wouldn't be much and could be considered negligible. Your best bet would be a massive, very bright star, not a sun-like star. A star like our sun that close would wreck havoc with the oort cloud and have a very high proper motion. $\endgroup$– userLTKSep 15, 2017 at 17:41
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$\begingroup$ "Your best bet would be a massive, very bright star, not a sun-like star" - How large/bright would a star have to be to have an apparent magnitude similar to the moon (or less), but be distant enough not to interfere with the Oort Cloud or cause significant differences in the gravity of the solar system? $\endgroup$– T. TraverSep 15, 2017 at 19:08
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The change in magnitude is given by: $$m_2-m_1=-2.5\ \log_{10}(B)$$
Where B is the change in brightness.
Alpha Centauri is about 4.4ly so inverse square says it would be 19.4x brighter at 1ly, so a magnitude differential of -3.2.
Its apparent magnitude would therefore go from -0.27 to -3.47. This is brighter than Sirius (-1.46) but not close to the Sun (-26.7) nor Venus. Venus doesn't cast a shadow, so neither would Alpha Centauri at 1ly.
On gravity, Jupiter is roughly one thousandth the mass of Alpha Centauri but would be about 63,000x closer than the star at 1ly. It would therefore have a gravitational effect on the Sun of: $$63\,000^2/1000=\mathrm{4\ million\ times}$$ that of Alpha Centauri at 1ly!
Sounds like this works for your story.
Data brightness sources: https://www.space.com/21640-star-luminosity-and-magnitude.html
Mass and distance sources: https://en.wikipedia.org
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1$\begingroup$ "Venus doesn't cast a shadow" - it depends. At a site with a pretty dark sky (low light pollution) Venus can definitely cast shadows. $\endgroup$ Sep 15, 2017 at 18:20
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1$\begingroup$ Just to add: at 1 ly separation, the stars would very likely not be bound gravitationally. Binary stars with some of the longest periods are separated by much smaller distances. astrophysicsspectator.com/topics/stars/BinaryStars.html $\endgroup$ Sep 15, 2017 at 19:02
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$\begingroup$ All light sources cast shadows. Even the CMB. $\endgroup$ Sep 16, 2017 at 10:23