How can I safely brighten my secondary star?

I think 20 AU is a reasonable distance (especially using a K type star). This would already get the apparent brightness of the second sun to around -20.5; taking 10 AU as the lower limit will get to -22 but given the next paragraph that red giant will probably disrupt the orbits, making 20 AU a safer lower bound.

However, we can go further. By replacing the second sun with a red giant that is both about the same mass as and much larger and brighter than the planet's primary, the second sun can be easily 20000x as luminous as the Sun. This would bring it up to -32 (!) but that's about 100x brighter than the sun and increase the temperature of the planet by over 3x.

However, choosing a point in the middle of the red giant phase can reach essentially any value in between -20 and -30; I suggest using something in the -22 to -24 range; but something brighter is fine as long as the planetary habitable zone is adjusted (which might need to affect the orbital distance of the second sun, etc.)

What if the other star was not all by itself, but had its light magnified? One way to do that is to put it in a reflection nebula. Reflection nebulae are clouds of dust that reflect light from a star embedded with the nebula. They're often found around young, hot stars, but it's not implausible to have one exist around a dimmer, cooler star.

In a 1922 paper, Edwin Hubble found that the relationship between a reflection nebula's apparent magnitude and its angular diameter, $R$, is $$5 \log_{10} (R) = -m + k$$ where $k$ is a constant resulting from a measurement. If we say that $k = 0$ (ideally), then we can calculate $R$. Assuming $m=-22$, that gives $R\approx10^{17}$, which is a pretty nonsensical answer. Why? Well, Hubble's relation was empirical, and holds for nebulae that are far away. I originally used it for some calculations; that was misguided. Realistically, a radius $r_0$ of about 1 light-year is reasonable, placing both stars inside the nebula.

But what if you could scale it down?

Sobolev (1960) did some modeling of scattering nebulae; it's a paper I've only just come across. He came up with one key equation, the ratio of the luminosity of the nebula to the luminosity of the star illuminating it: $$\frac{L_N}{L_*}=\frac{16\pi^2 r_0^2\bar{H}(\tau_0)}{L_*}$$ where $r_0$ is the radius of the (spherical-ish) nebula, $\tau_0$ is the optical depth at $r_0$ (and the optical depth isequals $\tau=\alpha r$ for some $\alpha$), and $$\bar{H}(\tau)=\frac{A}{\tau^2}\left(1-e^{-\tau}\right),\quad A\equiv\frac{L_*\alpha^2}{16\pi^2}$$ Putting this all together gives us $$\frac{L_N}{L_*}=\frac{16\pi^2r_0^2}{L_*}\frac{L_*\alpha^2}{16\pi^2}\frac{1}{\tau_0^2}\left(1-e^{-\tau_0}\right)=1-e^{-\tau_0}$$ A reasonable surface optical depth - and this is largely guesswork on my part - could be $\tau_0\approx2/3$, meaning the nebula is on the boundary of being optically thin or optically thick; we actually use $\tau=2/3$ to define the surface of the Sun. Therefore, we find $L_N\approx0.49L_*$, which is significant - assuming that my guess was anywhere near correct.

Is this plausible? You were correct in your comment that most nebulae are extremely large. This leaves us with two plausible possibilities:

1. The entire system is embedded in a nebula.
2. The nebula is somehow very small.

The first one would most likely produce some interesting but perhaps undesirable effects; for now, I'll consider that a non-viable option. (Also, the estimates above assumed that the other star is outside the nebula).

The second option, therefore, is the only one available. The problem is, it's tricky. Now, there are some nebulae that are small enough - on the order of 1 AU in diameter. If you want to explain it away by stating that, then you're fine. Another interesting option is to actively explain the small size by stating that stellar winds from the planet's primary star dissipated the originally much larger nebula in that part of the system, leaving it mainly only around the secondary star. I don't know what the timescales are there, although I can try to figure it out.

Stellar engineering is going to be quite difficult and expensive, but there are a few ways to increase the apparent light output of the secondary star in the binary system if the civilization is advanced enough.

The first and possibly easiest approach would be to simply focus the incoming solar radiation from the distant star. Platoons of mirrors or fresnel lenses in orbit can be used to amplify the incoming light, although the orbital mechanics of this would be rather tricky given the huge differential between the planet orbiting the primary sun and the 700 year orbit of the secondary. Given the rather faint amount of light being focused, I doubt that any evil genius will be able to focus the light of the distant secondary sun into a dangerous spot to burn cities and crops (the real danger would be hijacking the orbital arrays and refocusing the light of the primary). This would require a civilization with spacefaring technology at least equal to that of the 1960's on Earth (Saturn V class boosters might be required to place mirrors in very distant orbits, depending on the chosen arrangements of the orbital arrays, and possibly landing on other moons or planets to gather materials or establish control stations for the mirror array).

Given a higher level of technology, the secondary sun itself can be tweaked to deliver more energy to the planet. This would involve actually going to the distant sun and establishing a "solar laser". The mechanics of such a laser can be found here: http://laserstars.org/amateur/scifi.html, with the end result being the energy of the sun is focused into a beam and used to illuminate the distant planet. Since the laser light is monochromatic, the lighting will be a bit odd, but using something like yellow laser light should be acceptable to the inhabitants. Once again there will be issues with tracking the planet and keeping the beam placed on target, and there will be times when the primary sun is between the secondary sun and the planet, which might either result in the cutoff of the beam, or retargeting to secondary mirrors around the solar system to bounce the light around the sun and back to the planet. This would be possible using a somewhat more advanced technology than we have today (maybe two generations of technology beyond ours to get started on the process).

Getting into magitech, a truly advanced civilization would have some means of increasing the rate of fusion in the secondary sun to increase the luminosity. Since the rate of energy release is conditional on the amount of gravitational pressure in the core, there must be some means of "squeezing" the core, or perhaps introducing a "small" black hole into the stellar core to "pull" harder on the sun's core. The limitations of that (either external squeezing or using a black hole) is the increasing release of energy will push out on the material in the star, causing it to expand and cool unless overcome in some way (stars exist in a state of equilibrium between radiation pressure pushing outwards and gravitational forces pulling inwards). This sort of magitech engineering will probably require real time monitoring and active control over the process, keeping a crew of people or AI's in close proximity to the star in order to regulate the process. This is obviously well beyond any current or projected capabilities of our current civilization, but perhaps in a few centuries we will be able to carry out a project of that nature.

The other very long term possibility would be to physically bring the distant sun closer to the planet (from 100 AU to 10 AU). Moving planets, even giant planets is possible for the very patient (slingshotting asteroids past a planet and using momentum exchange, much like spacecraft slingshot past giant planets to change orbits and speed up or slow down), but since stars are orders of magnitude larger than even giant planets, getting enough momentum exchange to physically move a star is going to take either an unreasonably long time, or require vast amounts of energy (a stream of asteroids moving at a significant fraction of c might do the trick, but the energy needed to do so would be a large fraction of the output of the entire star). Using the energy output of the star to create a sort of rocket (using a truly enormous array of mirrors to reflect the star's energy back on a spot to create an intense "flare" of plasma to provide thrust. This idea was explored in the SF Novel "Bowl of Heaven" and Shipstar" by Larry Niven and David Brin), but once again , we are talking very long term.

Roughly, to get the brightness from that of the full moon to an overcast midday, you need to increase the luminosity about 4000 times (Wikipedia: Daylight). Thus you would have to bring in the secondary star from 100 AU to 10 AU to get it 100 times brighter (i.e. from 40 times the brightness of the moon to 4000 times the brightness of the moon). The orbit of your planet might actually be stable, but I'd be worried (hopefully someone writing here can do a simulation). Gamma Cephei has a large (1.5 solar mass) star and a red dwarf (0.5 solar mass) that are 20 AU apart, so there's a lot better chance of a planetary orbit around the heavy star being stable in Gamma Cephei. So you might have to increase the distant star's luminosity.

For instance, using the mass luminosity relationship you can move the companion to 20 AU and make it 1.4 times the mass of the sun. This will quadruple the actual luminosity which will balance being moved twice as far away. That would be near the greatest mass of an F star, so it would be pumping out a tad bit more ultraviolet.

Another approach might be to have much higher metallicity in the companion star (compared to the sun) which would also increase its luminosity, these are one kind of subgiant star. Or you could have the other kind of subgiant, which are old stars that have moved off the main sequence, but the latter type are slowly moving toward giant star status, which would be a bad, bad thing for your planet.

As I state below, if both stars in the system are suitable for having planets old enough to have interesting stuff like habitable biospheres, complex multi celled plants and animals, or native intelligent beings, etc., there will be only a narrow range of luminosity difference between them. The brighter star can be only about 5 or 6 times as bright as the dimmer star.

Assume, therefore, that the planet orbits a star that is not a G2V like the Sun but about a K5V, much less luminous than The Sun, and therefore the planet orbits much closer to that star and has a much shorter year. If the other star is something like a G0V or a F8V it should be somewhat brighter than the Sun and about 5 or 6 times as bright as the star the planet orbits. If the distance between the 2 stars is 10 times the distance between the planet and the dimmer star that it orbits. the apparent brightness of the farther and brighter star will be diminished by 100 times and thus it will appear only 5 or 6 percent as bright as the nearer star as seen from the planet.

Only 5 or 6 percent as bright as the other star is not very bright as compared to the other star, but on the other hand if the nearer star gives the planet about the same amount of light as the Sun sheds on Earth, 5 or six percent of that should equal about 20,000 to 24,000 times the brightness of the full moon on Earth!

You will have to find out if that will be enough for Humans to see colors, for movement to be easy and safe, for the sky to look blue and the stars to be masked by the sky's brightness, etc. I think it should be enough.

If the other star can come as close as five times the orbital radius of the planet around its star, then the other star could appear as bright as 0.2 to 0.24 as bright as the nearer star, or about 80,000 to 96,000 times the brightness of the full moon.

I believe the brightness of the full moon is give as 0.25 lux. 20,000 to 24,000 times the brightness of the full moon would be about 5000 to 6,000 lux, a few times the brightness of a typical overcast day:

1,000 - 2,000 lux Typical overcast day, midday

80,000 to 96,000 time the brightness of the full moon would be about 20,000 to 24,000 lux, equal to:

20,000 lux Shade illuminated by entire clear blue sky, midday

https://en.wikipedia.org/wiki/Daylight1

You should remember that the minimum distance between the stars for the planet's orbit to be stable is not a distance in Astronomical Units but a ratio of the distance between the planet and the star it orbits and the distance to the other star.

However, where the separation is significantly less, a stable orbit may be impossible. If a planet's distance to its primary exceeds about one fifth of the closest approach of the other star, orbital stability is not guaranteed.[62]

One study of Alpha Centauri, the nearest star system to the Sun, suggested that binaries need not be discounted in the search for habitable planets. Centauri A and B have an 11 AU distance at closest approach (23 AU mean), and both should have stable habitable zones. A study of long-term orbital stability for simulated planets within the system shows that planets within approximately three AU of either star may remain stable (i.e. the semi-major axis deviating by less than 5%). The HZ for Centauri A is conservatively estimated at 1.2 to 1.3 AU and Centauri B at 0.73 to 0.74"”well within the stable region in both cases.[5]

https://en.wikipedia.org/wiki/Habitability_of_binary_star_systems2

Wiegert, Paul A.; Holman, Matt J. (April 1997). "The stability of planets in the Alpha Centauri system". The Astronomical Journal. 113 (4): 1445"“1450. arXiv:astro-ph/9609106Freely accessible. Bibcode:1997AJ....113.1445W. doi:10.1086/118360.

However, where the separation is significantly less, a stable orbit may be impossible. If a planet's distance to its primary exceeds about one fifth of the closest approach of the other star, orbital stability is not guaranteed.[62]

https://en.wikipedia.org/wiki/Planetary_habitability#Binary_systems3

In binary star systems, however, a planet must not be located too far away from either one star or too close to two "home" stars or its orbit will be unstable. If that distance exceeds about one fifth of the closest approach of the other star, then the gravitational pull of that second star can disrupt the orbit of the planet (Graziani and Black, 1981; Pendleton and Black, 1983; and Dvorak et al, 1989).

http://www.solstation.com/habitable.htm4

If you consider it desirable for the farther star to be more than 0.25 times as bright on the planet's surface as the nearer star that the planet orbits, or if you want the star that the planet orbits to be as bright as the sun, then the other star will have to be too luminous to be old enough for its planets to be habitable for humans or have higher life forms. And since the two stars and all their planet should be the same age, the planet in question and the star that it orbits will also have to be too young for the planet to be interesting.

Unless scientists note that the planet should not yet be habitable or have advanced lifeforms. Thus characters may speculate that advanced aliens terraformed the planet, or that the entire planet was moved from an older solar system into this younger solar system by super advanced aliens. And maybe someone will point out that the clock is ticking and there are "only" a few million years left until the brighter star swells into a red giant and all life in the system is destroyed.

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My Previous Answer.

For scientific reasons I have reversed your star designations, making Star B the one that Planet X orbits and Star A the more distant star.

If planet X orbits Star B but not Star A, Star A should be at least ten times as far away from Planet X as Star B is, in order for the orbit of planet X to be stable. If this is supposed to be hard science fiction you will need a more expert opinion. Of course the distance between Star A and Star B can be many times the minimum of ten times the radius of Planet X's orbit around Star B.

If the distance between Star A and Star B is exactly 10 times the radius of Planet X's orbit around Star B, then some times Planet X will be exactly 11 times as far from Star A as from Star B. And sometimes Planet X will be only 9 times as far from star A as from Star B. The distance from Star A to Planet X will vary between 0.9 and 1.1 times the average distance.

And since the amount of light planet X receives from star A varies with the square of the distance, that amount will vary from 0.826 to 1.234 of the average amount.

If the distance between Star A and Star B is exactly 100 times the radius of Planet X's orbit around Star B, the amount of light that Planet X receives from Star A will vary between 0.980 and 1.019 of the average amount of light.

Since that is a smaller range of difference, as a general rule you would want the distances between Star A and Star B to be as many times greater as possible than the radius of the orbit of Planet X around star B.

But you also need the distance between Star A and Star B to be as small as possible compared to the radius of the orbit of Planet X around Star B. If you want Planet X to be interesting because it is habitable for Earth Humans, or has advanced multi celled life like trees and mammals, or has native intelligent beings.

If Star A is 10 times as far away from Planet X as Star B is, which I think is the minimum distance for Planet X to have a stable orbit, it will have to be 100 times as luminous as Star B to give Planet X as much light as Star B does. If Star A is only as luminous as Star B it will give Planet X only one percent of the light that Star B gives planet X.

If Star A is 100 times as far away from Planet X as Star B is, it will have to be 10,000 times as luminous as Star B to give Planet X as much light as Star B does. If Star A is only as luminous as Star B it will give Planet X only one hundredth of one percent (or 0.0001) of the light that Star B gives planet X.

You didn't specify the desired ratio between the apparent brightness of Star A and Star B as seen from Planet X. You just said Star A (your Star B) should give Planet X enough light to make a difference. And you didn't specify whether you meant enough light to make a difference in the temperature of Planet X or merely enough light to make a difference in it's illumination.

If you want Star A to shed as much light on Planet X as Star B does, then the ratio of their relative absolute luminosity must equal the square of the ratio of their relative distances from Planet X. If Star A is 10 times as distant as Star B it will have to be 100 times as luminous to appear exactly as bright in the sky of Planet X. If Star A is 100 times as distant as Star B it will have to be 10,000 times as luminous to appear exactly as bright in the sky of Planet X. If Star A is 1,000 times as distant as Star B it will have to be 1,000,000 times as luminous to appear exactly as bright in the sky of Planet X.

Thus if Star A and Star B have to have anything remotely resembling the same brightness in the sky of planet X, Star A should have at least several times the absolute luminosity of Star B, and possibly up to millions of times the luminosity. Thus Star A would be much intrinsically brighter than Star B. Thus Astronomers would call it A and call the star that Planet X orbits B. Because of the high probability that the more distant star would be more luminous than the star Planet X orbits, I switched the designations of the stars from what they were in your question.

Suppose that you desire star A to appear 0.0001 times as bright in the sky of Planet X as Star B. Then if Star A is 0.10 times as luminous as Star B, and 10 times as far from Planet X, it will appear to be 0.0001 times as bright from the surface of Planet X. If Star A is exactly as luminous as Star B, and 100 times as far from Planet X, it will appear to be 0.0001 times as bright from the surface of Planet X. If Star A is 1,000 times as luminous as Star B, and 1,000 times as far from Planet X, it will appear to be 0.0001 times as bright from the surface of Planet X. If Star A is 10,000 times as luminous as Star B, and 10,000 times as far from Planet X, it will appear to be 0.0001 times as bright from the surface of Planet X.

Thus even if Star A appears only 0.0001 times as bright as Star B as seen from Planet X, it could, depending on its distance, be tens, hundreds, or even many thousands of times as absolutely luminous as Star B, the star that Planet X orbits.

By comparison, the Sun has an apparent brightness as seen from Earth 398,110 times as bright as the apparent brightness of an average full moon. The apparent brightness of the full moon is 0.0000025 that of the Sun, so if Star B appears as Bright as the Sun from Planet X and Star A appears only 0.0001 as bright as star B as seen from Planet X that could still be about 40 times as bright as a full moon seen from Earth.

The absolutely most luminous star known to science is R136a1 in the Large Magellanic Cloud, about 8,710,000 times as luminous as the Sun. The least luminous known star is 2MASS J0523-1403, about 0.000126 times as luminous as the Sun. That gives a luminosity range of about 69,126,983,000 times. That should be enough for any desired difference in the luminosity of the two stars in the solar system of Planet X, right?

Wrong.

If you want Planet X to be interesting because it is habitable for Earth Humans, or has advanced multi celled life like trees and mammals, or has native intelligent beings, Planet X must have enjoyed a relatively constant amount of radiation from it's sun, Star B, for billions of years, since Earth is believed to be relatively typical, and it took billions of years for those things to develop on Earth.

Therefore Star B that Planet X orbits must have been a relatively stable main sequence star for billions of years in order for Planet X to be habitable for Earth Humans, or have advanced multi celled life like trees and mammals, or have native intelligent beings. And since both stars in the system would be the same age, Star A must also have been a relatively stable main sequence star for billions of years. When stars eventually leave the main sequence they usually change in ways that destroys all life on the planets that orbit them and may also destroy all life on planets orbiting other stars in the same star system.

And what types of stars will remain stable main sequence stars for billions of years? Stars of late spectral type F (starting at maybe type F8), type G, Type K, and type M. Thus Star B, that Planet X orbits, and Star A, in the same star system, would both have to be somewhere between about spectral type F8V to M9V, which would limit the possible range of their luminosity difference. I believe the extreme possible luminosity difference between Star A and Star B would be about 25 times.

But many scientists believe that stars from mid type K and all type M stars are not suitable for having habitable planets for various reasons. If that is correct the possible spectral types for Star B would be limited to about F8V to K5V. That gives a luminosity range of about six times for the difference between Star A and Star B. But since it is not specified whether Star A should have any habitable planets its spectral type can be between type F8V and type M9V.

So if you want your story to be anything like hard science fiction you should find more precise figures for the various limits listed before making your calculations, if you want Planet X to be interesting because it is habitable for Earth Humans, or has advanced multi celled life like trees and mammals, or has native intelligent beings. Unless the stars in the star system are younger and should not have planets as advanced as Planet X seems to be. Perhaps super powerful aliens terraformed Planet X millions of years ago and seeded it with life forms billions of years more advanced than it had time to evolve naturally, or even took Planet X from its original star system and moved it into the much younger star system it is now in.