# Are there stars that orbit other stars, such that the barycenter is inside one of them?

I know binary stars exist but I mean a star which orbits with a barycenter that is inside another star, much like a moon. Why is such a system so rare?

• What is so special about stars? Why not consider 'objects' that orbit each other? Like stars orbiting around a massive black hole, or black holes orbiting each other? Or galaxies orbiting each other. Or galaxy systems orbiting each other...... Dec 21, 2023 at 18:03
• @Roland each of those situations would require a completely different answer. So from that point of view, stars are special - they suffer tidal distortion, the orbits don't lose significant gravitational wave losses and they don't consist of multiple components. Dec 22, 2023 at 7:14

Such systems are rare, but not impossible, and I explain why below. An example can be found in Stevens et al. (2020). This consists of a 3.3 solar mass B star orbited by a 0.22 solar mass M-star in a 3.62 day period, eclipsing binary sytem, where the components are separated by about 14.8 solar radii. The centre of mass is only (see below for formulae) 0.9 solar radii from the center of the more massive star, which itself has a radius of about 3.1 solar radii. Thus the barycenter of the system is inside the massive star.

Consider a binary system with star centers separated by $$a$$ and barycentre at the origin. The positions of the individual stars is given by $$m_1 a_1 = m_2 a_2\ ,$$ where $$a_1+a_2= a$$. Thus $$a_1 = \frac{m_2}{m_1+m_2}a$$ and for the situation of a barycenter inside the more massive star ($$m_1$$), we need $$a_1 < r_1$$.

For this to occur then, we need two conditions - the more massive star needs to be (much) more massive, so that $$m_2 \ll m_1 + m_2$$ and the system needs to be as compact as possible so that $$a$$ is not too much bigger than $$r_1+r_2$$.

A concrete example might help. Suppose the radius of a main sequence star (in solar units) is roughly equal to its mass (in solar units). We could imagine a solar type star and a 0.1 solar mass M-dwarf. In order for the barycentre to be inside the solar type star then $$\frac{0.1}{1.1}a < 1$$ and so $$a< 11$$ solar radii and an orbital period of less than 4 days. If the mass ratio was less extreme then the objects would have to be even closer.

However, there is a snag if they get too close, and that is Roche lobe overflow (RLOF). If stars get too close together then the system becomes unstable to mass transfer as the larger star fills its Roche Lobe. For a given separation, the value of $$a_1$$ at which this occurs will be larger for larger mass ratios (i.e. when the two stars are more similar).

The result of RLOF in this situation is likely to be a W UMa system, or "contact binary", where the two stars are quickly brought even closer together and can no longer be separated.

So that is why the situation you propose is rare. Contact binaries are relatively common and arise when two stars orbit too close to each other. However, detached binary systems where the barycenter is inside one of the stars requires a large mass ratio (reasonably rare) and a close separation (rare) but not too close.

(exo)-Planets are (almost by definition) much smaller (less mass) than stars. So in a system with a (solitary) star and planets, the mass of the star will dominate. It is quite possible that the barycentre of a planet and a star will be inside the star. This isn't always the case, and the barycenter of the solar system is sometimes inside, sometimes outside the sun (mostly depending on the relative positions of Jupiter and Saturn)

The mass ratio of the Sun to Jupiter is 1000:1 — a big difference.

Stars, however, are all "star-sized"(!) So there is at least a fair chance that the stars in a binary system will be of similar size, or at least that the difference in sizes is such that the barycentre is outside the larger star. For example, α Oph is a star system with quite a big difference in sizes. The main star is a bright white star with a mass of 2.4 times the sun. The secondary star is a fairly dim orange dwarf, it's mass is 0.85 of the sun. So you see that the mass ratio is about 3:1, not so much of a difference.

Moreover, stars can orbit at a very great distance from each other, and the barycentre is more likely to be inside the primary when the two stars are close.

If there is to be a big difference in mass, then the bigger star must be very massive, and such stars are very rare. Pairs of stars with a massive primary and tiny secondary are rarer still, and finding those with very tight orbits is difficult. I've not been able to identify such a system (ignoring exceptional cases like contact binaries)

But there is nothing special about the barycentre being inside the star. As I noted, the barycentre of the solar system is often not inside the sun. Yet we still say that the planets orbit the sun.

• There is a case not considered: contact binaries. The barycenter of such a system is definitely inside one star, possibly both simltaneously? I guess that would depend on how you define the boundary of each star in a contact binary. Dec 21, 2023 at 2:04
• considered, but I dont think thats what the OP wants Dec 21, 2023 at 2:13