An analytic framework modeling the gravitational environment of stars orbiting a galactic supermassive black hole

1. Introduction

The gravitational environment at the center of a galaxy is extraordinary. The gravitational influence due to the galactic supermassive black hole (SMBH) plus a high density of dark matter has set up an exotic environment for stars at the deep galactic center. Moreover, the SMBH and dark matter are not independent individual components. Simulations suggest that the adiabatic growth of a galactic SMBH would re-distribute the surrounding dark matter density to form a density spike under the standard cold dark matter paradigm.¹ This theoretical prediction has been supported by some recent studies of SMBH and stellar-mass black holes.^2,3 The dark matter density spike surrounding a galactic SMBH can provide some interesting gravitational effects on the nearby stars. In fact, some previous studies have already considered the effects of dark matter or different mass distributions on the S2 star.^{4,5,6,7,8,9,10} Specifically, different models of ultralight dark matter^8,9,10 and keV fermion darkinos⁵ are examined and constrained. However, these dark matter models considered are generally different from the dark matter density spike model suggested in Ref. 1.

In this paper, based on the previous results of the S2 star analyses in Refs. 4, 6 and 7, I follow the dark matter density spike model in Ref. 1 and formulate an analytic framework to combine the gravitational effect of the dark matter density spike and SMBH. I outline some intriguing features of stars orbiting the SMBH (not only limited to the S2 star) and show that the orbital precession and orbital shrinking rates can be specifically determined by the SMBH mass. This provides the first analytic framework to describe the gravitational environment of stars orbiting a galactic SMBH in the standard cold dark matter paradigm.

2. The Analytic Framework

By using the data of 43 galaxies, a study has revealed an empirical relation between the SMBH mass and the total dynamical mass of galaxies $M_{\mathrm{\text{tot}}}$¹¹ :

This empirical relation agrees with the later simulation result $M_{\mathrm{\text{BH}}}\propto M_{\mathrm{\text{tot}}}^{1.55\pm 0.05}$¹² and the results for elliptical galaxies $M_{\mathrm{\text{BH}}}\propto M_{\mathrm{\text{tot}}}^{1.6}$.¹³ Therefore, the SMBH mass is a good indicator to probe the total mass of a galaxy.

On the other hand, using the theory of cosmological structure formation, the concentration parameter $c_{200}$, which measures the “concentration of dark matter in a galaxy”, can be determined by the total mass of a galaxy. Based on the data of galaxies and galaxy clusters, one can write the empirical relation as¹⁴

where $h\approx 0.7$ is the Hubble parameter, $C_0=5.119_{-0.185}^{+0.183}$, ${\gamma }_c=0.205_{-0.010}^{+0.010}$ and ${log}_{10}(M_0/M_{\odot })=14.083_{-0.133}^{+0.130}$. Apart from the concentration parameter, we can also relate the virial radius of a galaxy with the galactic total mass by $r_{200}^3=3M_{\mathrm{\text{tot}}}/4\pi {\rho }_{200}$, where ${\rho }_{200}$ is defined as ${\rho }_{200}=200\times (3H_0^2/8\pi G)$, with $H_0\approx 70$km/Mpc/s being the Hubble constant.

If dark matter particles are collisionless, numerical simulations show that the density profile of dark matter particles in a galaxy would follow the Navarro–Frenk–White (NFW) density profile¹⁵ :

where ${\rho }_s$ is the scale density and $r_s=r_{200}/c_{200}$ is the scale radius. This profile is commonly modeled as the galactic dark matter density profile in many previous studies.¹⁶ The scale density can be obtained from the galactic total mass : The NFW scale density ${\rho }_s$ and scale radius $r_s$ can characterize the dark matter distribution of a galaxy and determine the general gravitational properties of a galaxy. Based on the above formulation, since $M_{\mathrm{\text{BH}}}$ determines $M_{\mathrm{\text{tot}}}$ and $M_{\mathrm{\text{tot}}}$ determines both ${\rho }_s$ and $r_s$, we can see that ${\rho }_s$ and $r_s$ can be specifically determined by the SMBH mass $M_{\mathrm{\text{BH}}}$.

Now we turn our focus to the dark matter density spike model near the SMBH. If the SMBH grows adiabatically, the original NFW dark matter density function near the SMBH would be altered to form a dark matter density spike due to conservation of angular momentum and radial action.¹ Outside the spike region $r\ge r_{\mathrm{\text{sp}}}$, the dark matter density function would follow back to the NFW density profile. To summarize, the dark matter density around the galactic SMBH can be described by the following spike model (with general relativistic correction)³ :

where $R_s=2G{M}_{\mathrm{\text{BH}}}/c^2$. Moreover, the adiabatic benchmark model can relate the spike index ${\gamma }_{\mathrm{\text{sp}}}$ with the power-law index of the dark matter density outside the spike region $\gamma$: ${\gamma }_{\mathrm{\text{sp}}}=(9-2\gamma )/(4-\gamma )$.¹ As $\gamma =1$ for the NFW density profile, the adiabatic growth model predicts ${\gamma }_{\mathrm{\text{sp}}}=7/3\approx 2.33$. However, if stellar heating is significant, the spike index could be as small as ${\gamma }_{\mathrm{\text{sp}}}\approx 1.5$.¹⁷ Therefore, the spike index generally ranges from ${\gamma }_{\mathrm{\text{sp}}}=1.5$ to ${\gamma }_{\mathrm{\text{sp}}}=2.33$.

The spike radius $r_{\mathrm{\text{sp}}}$ in the standard dark matter density spike model is empirically defined by $r_{\mathrm{\text{sp}}}=0.2r_{\mathrm{\text{in}}}$, where $r_{\mathrm{\text{in}}}$ is the radius of influence.³ The radius of influence can be determined by¹⁸

Therefore, one can relate $r_{\mathrm{\text{sp}}}$ with the SMBH mass as Also, by considering the dark matter density at $r=r_{\mathrm{\text{sp}}}$ in Eq. (5), we can get Hence, the above two relation equations (7) and (8) can connect ${\rho }_{\mathrm{\text{sp}}}$ and $r_{\mathrm{\text{sp}}}$ individually with $M_{\mathrm{\text{BH}}}$ and the spike index ${\gamma }_{\mathrm{\text{sp}}}$. If we take the benchmark value ${\gamma }_{\mathrm{\text{sp}}}=2.33$ as evidenced in a recent study,³ using the SMBH mass can sufficiently determine the values of ${\rho }_{\mathrm{\text{sp}}}$ and $r_{\mathrm{\text{sp}}}$. These values can constrain the gravitational and dynamical behavior of a star orbiting the SMBH. Moreover, one can find a power-law relation between $M_{\mathrm{\text{BH}}}$ and ${\rho }_{\mathrm{\text{sp}}}{r}_{\mathrm{\text{sp}}}^{{\gamma }_{\mathrm{\text{sp}}}}$ (with a correlation coefficient $R^2>0.99$): $M_{\mathrm{\text{BH}}}=K{({\rho }_{\mathrm{\text{sp}}}{r}_{\mathrm{\text{sp}}}^{{\gamma }_{\mathrm{\text{sp}}}})}^{\beta }$, where K is the proportionality constant and $\beta$ is the correlation power-law index (see Table 1 for the corresponding values). This relation can easily relate the dark matter gravitational effect to SMBH mass directly.

3. Gravitational Signatures Around a SMBH

The gravitational influence due to a galactic SMBH plus the dark matter density spike can produce intriguing effects on the stars orbiting the SMBH. In the following, we will specifically discuss two prominent signatures.

3.1. Orbital precession

For stars orbiting the galactic SMBH, the orbits would undergo small precession due to the general relativistic effect. This can be observed from the data of S2 star orbiting the SMBH in our Milky Way, which agrees with the general relativistic prediction.⁴ However, with the addition of the dark matter density spike, the situation could be more complicated.

Following general relativity, for a star orbiting the SMBH inside the dark matter density spike, its motion on the fixed plane can be given by⁷

where $u=1/r$, L is the angular momentum, and $M=M_{\mathrm{\text{BH}}}+M_{\mathrm{\text{DM}}}$ is the total enclosed mass, with $M_{\mathrm{\text{DM}}}\approx 4\pi {\rho }_{\mathrm{\text{sp}}}{r}_{\mathrm{\text{sp}}}^{{\gamma }_{\mathrm{\text{sp}}}}{r}^{3-{\gamma }_{\mathrm{\text{sp}}}}/(3-{\gamma }_{\mathrm{\text{sp}}})=4\pi K{M}_{\mathrm{\text{BH}}}^{\beta }{r}^{3-{\gamma }_{\mathrm{\text{sp}}}}/(3-{\gamma }_{\mathrm{\text{sp}}})$ being the enclosed mass of dark matter. Let $u=u_0+\mathrm{\Delta }u$ with $\mathrm{\Delta }u\ll u_0$. By expanding all terms up to the first order of $(\mathrm{\Delta }u/u_0)$, we get The above relation is a very good approximation if the orbital eccentricity is small.

By choosing the arbitrary $u_0$ such that the first term on the right-hand side of Eq. (10) is zero, we get

where For a very small orbital precession angle, we have $u_0\approx G{M}_{\mathrm{\text{BH}}}/L^2\approx {[a(1-e^2)]}^{-1}$, where a and e are the semi-major axis and eccentricity of the stellar orbit, respectively. This gives the analytic form of the precession angle per one period : Therefore, the precession angle can be determined by the SMBH mass analytically. One interesting signature is that the precession term due to general relativity $6\pi G{M}_{\mathrm{\text{BH}}}/a(1-e^2)c^2$ is prograde in direction (positive contribution to $\mathrm{\Delta }\varphi$) while the precession term due to the dark matter density spike $-4{\pi }^{2}K{M}_{\mathrm{\text{BH}}}^{\beta -1}{[a(1-e^2)]}^{3-{\gamma }_{\mathrm{\text{sp}}}}$ is retrograde in direction (negative contribution to $\mathrm{\Delta }\varphi$). Hence, there exists a critical position (i.e. a particular value of the semi-major axis) such that the prograde precession balances the retrograde precession (i.e. no precession) :

3.2. Orbital shrinking

For a star with mass $m\ll M_{\mathrm{\text{BH}}}$ orbiting the galactic SMBH, some energy would be lost due to gravitational wave emission. For small orbital eccentricity e, the energy loss rate can be approximately given by³

Moreover, when the star is moving through the dark matter density spike, the gravitational effect of the star would pull the dark matter particles toward it so that a slightly higher concentration of the dark matter particles would be located behind the moving star. This exerts a collective gravitational force slowing down the star, which is known as dynamical friction. For small orbital eccentricity, the energy loss rate originated from dynamical friction due to the dark matter density spike is given by³ where $ln\mathrm{\Lambda }\approx ln\sqrt{M_{\mathrm{\text{BH}}}/m}$ is the Coulomb Logarithm,¹⁹$\xi (\sigma )\approx 1$ is a numerical factor depending on the dark matter velocity dispersion $\sigma$,³ and $v\approx \sqrt{G{M}_{\mathrm{\text{BH}}}/a}$. The total energy loss rate of the stellar orbital motion is $Ė={Ė}_{\mathrm{\text{GW}}}+{Ė}_{\mathrm{\text{DF}}}$.

Consider the mean orbital radius a being much greater than $R_s$ so that the general relativistic effect is negligible. Based on the Keplerian relation, the orbital shrinking rate is

where $E=-G{M}_{\mathrm{\text{BH}}}m/2a$. Since the dark matter density ${\rho }_{\mathrm{\text{DM}}}={\rho }_{\mathrm{\text{sp}}}{(r_{\mathrm{\text{sp}}}/a)}^{{\gamma }_{\mathrm{\text{sp}}}}=K{M}_{\mathrm{\text{BH}}}^{\beta }{a}^{-{\gamma }_{\mathrm{\text{sp}}}}$ depends on $M_{\mathrm{\text{BH}}}$, one can relate the orbital shrinking rate $ȧ/a$ with $M_{\mathrm{\text{BH}}}$ for different a and m. By defining the time scale of orbital shrinking as $t_s=a/ȧ$ (i.e. the approximate time for the star falling into the SMBH completely) and taking $m\sim M_{\odot }$, we can relate $t_s$ with a for different $M_{\mathrm{\text{BH}}}$. As $t_s$ generally increases with a, there exists a threshold value of $a_{min}$ such that the orbital shrinking time scale $t_s$ is smaller than the age of galaxy formation ($\sim 13$ Gyr). Hence, for any solar mass star with $a\le a_{\mathrm{\text{min}}}$, it would be disrupted by the SMBH within the age of galaxy formation. The threshold mean orbital radius $a_{min}$ can be regarded as the ‘disruption radius’ within the time $t_s$. From Eqs. (15)–(17), we can get the relation $a_{min}$ as a function of $M_{\mathrm{\text{BH}}}$, assuming the benchmark value of ${\gamma }_{\mathrm{\text{sp}}}=2.33$ (see Fig. 1). Also, as seen from Fig. 1, there exists a critical $M_{\mathrm{\text{BH,c}}}$ such that $a_{min}$ would be the smallest. It is because the energy loss rate due to dynamical friction dominates when $M_{\mathrm{\text{BH}}}\le M_{\mathrm{\text{BH,c}}}$ while the energy loss rate due to gravitational wave emission dominates when $M_{\mathrm{\text{BH}}}\ge M_{\mathrm{\text{BH,c}}}$. Therefore, we expect that a smaller region of stars would be disrupted if the galactic SMBH mass $M_{\mathrm{\text{BH}}}\approx M_{\mathrm{\text{BH,c}}}$. In the regime when dynamical friction dominates the total energy loss rate, we can get $a_{min}^{{\gamma }_{\mathrm{\text{sp}}}-3/2}\propto M_{\mathrm{\text{BH}}}^{\beta -3/2}ln{M}_{\mathrm{\text{BH}}}$ with a fixed $t_s$. Since $\beta \le 3/2$ and ${\gamma }_{\mathrm{\text{sp}}}\ge 3/2$, a smaller SMBH mass can generally give a larger disruption radius $a_{min}$.

4. Discussion and Conclusion

In this paper, I present an analytic framework to describe the gravitational environment surrounding a galactic SMBH. In particular, we have discussed two intriguing features of stars orbiting the galactic SMBH: the stellar orbital precession and orbital shrinking. These features and quantified effects can be uniquely determined by the SMBH mass. Given that galactic SMBH mass can be effectively determined by X-ray observations, the gravitational features surrounding different galactic SMBHs could be generally described and predicted by this analytic framework.

Moreover, future observations of the S-star cluster orbiting the SMBH in the Milky Way center (i.e. Sgr A*) can be another useful way to verify our described picture. For example, the short-period S4716 star (with orbital period 4 years)²⁰ and the long-period S12 star (with orbital year 59 years)²¹ might be able to describe different gravitational environment of Sgr A*. Due to the short orbital period of the S4716 star, we can determine the orbital precession angle relatively easier. For the S12 star, it has already been monitored for nearly 30 years and it has a long orbital semi-major axis for revealing the gravitational influence at a longer distance from the Sgr A*. Therefore, by combining these data, we can get a more comprehensive understanding of the gravitational environment surrounding a SMBH. Besides, future low-frequency gravitational wave observations (e.g. LISA²²) can also provide a new window to illustrate the possible orbital shrinking due to gravitational wave emission and dynamical friction of dark matter.

ORCID

Man Ho Chan  https://orcid.org/0000-0001-5088-9117