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I read about one of the stars orbiting Sagittarius A* and according to Wikipedia, it reaches a maximum speed of 0.03c during its orbit. Would this (or any other factor) make it impossible for a planet to have a stable orbit around this star?

https://en.wikipedia.org/wiki/S2_(star)

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Technically yes, but no.

Since the Wikipedia article regarding S2 notes an orbital period of 16.0518 years, with an eccentricity of 0.88466. A typical B0V star has a mass of ~18 solar masses while Sgr A* has a mass of 4.1 million solar masses. Applying Kepler's third law to Sgr A* and S2: $$\sqrt[3]{\frac{(16.0518 \text{ yr})^2 \cdot G(4.1\cdot10^6 M_\odot+18M_\odot)}{4\pi^2}} \approx 1014 \ \text{AU}$$

The Hill sphere formula is $r_H = a(1-e)\sqrt[3]{\frac{m}{3M}}$. Plugging in the masses, semi-major axis, and eccentricity in, we get $\sim 1.328 \text{ AU}$ as the Hill sphere. While objects orbiting at the edge of the Hill sphere may seem like they are stable, but long-term stability tends to indicate that objects orbiting at half of the sphere are stable, meaning that a planet could stably orbit at 0.5 AU.

That said, a planet probably could not form around this star (see Why don't we detect planets around OB stars and no terrestrial planets around A or early F stars?). Furthermore, like @JamesK said in his great answer, such as star's strong stellar winds would have blown away almost all its disk material early, so planets would not have time to form.

Assuming a bond albedo of 0.1 for our hypothesized rocky planet and a luminosity of $50000 L_\odot$ for S2, we have $T^4=1.08586\cdot10^{15} K^4$. Taking this to the fourth root gives 5740 Kelvin. This is way too hot - almost as hot as the Sun, which is enough to boil away the crust, mantle, and whatever is left of the planet. Therefore, while orbits could be stable, no planet could exist due to the tiny orbital radius, extreme temperatures, and unsuitable formation conditions.

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  • $\begingroup$ The body of the question asks about the specific star S2, the title asks about stars like S2 near the galactic core. Stars like S2 in being B0V stars would be too hot for close planets. Stars like S2 in being near to the galactic black hole, but much less massive and several times as far away, which would still be really close, could possibly form and keep planets. $\endgroup$ Commented Dec 19, 2021 at 16:45
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In principle such a star could have a planet. Although the star is moving fast, a planet would also be pulled by the black hole's gravity, so you only need to consider tidal effects, and the "Hill sphere". If the planet orbits in the Hill sphere, it can orbit safely.

A planet orbiting a 4 million solar mass black hole would at 120 AU would have a Hill sphere with a radius of about 1 AU. So planets would need to be nearer to the start than the Earth is to the sun to be in stable orbits. This is possible. But note that S2 is a massive hot star. Any planet at that radial distance would be cooked!

It may also be that the powerful solar wind of a young 15 solar-mass star will clear material further than 1 AU, making it hard to form planets that close to the star.

I've also not taken relativity into account, nor the elliptical nature of the orbit of S2, nor do I need to. Instead I've just used the periapse distance and Newtonian gravity. These should allow for a reasonable estimate of the Hill radius. For longer term stability one should be conservative. So if the planet orbits at 0.5 au it will surely be stable, but even more cooked!

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    $\begingroup$ The periapsis distance of S2 is ~1400 times the Schwarzschild radius of Sag A*, so a Kepler orbit's a reasonable 1st approximation, both for it and any planets it may have. Wikipedia says that its Schwarzschild precession is ~12 arc-minutes; I presume that's per orbit. $\endgroup$
    – PM 2Ring
    Commented Dec 19, 2021 at 1:17

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