Doppler Boosting of the S-stars in the Galactic Center

S-stars are a population of young, bright stars orbiting within 0.04 pc from the supermassive black hole (BH) in the Galactic Center (Habibi et al., 2017). Long-term monitoring of these stars reveal their Keplerian motion around the central BH with periods as short as tens of years and, in many cases, high eccentricities (Ghez et al., 2003a; Gillessen et al., 2017). Observations of one particular member of the S-cluster — the relatively bright (m${}_{K}\approx 14$) star S2 making a full orbit in about 16 yr and approaching the BH as close as $115$ au (Schödel et al., 2002; Ghez et al., 2003b) — allowed accurate measurements of the BH mass and the distance to the Galactic Center (Ghez et al., 2000; Eisenhauer et al., 2003), as well as the detection of some post-Newtonian effects, namely the gravitational redshift (Gravity Collaboration et al., 2018; Do et al., 2019a) and general relativistic (GR) precession (Gravity Collaboration et al., 2020a). Recently, several fainter S-stars approaching Sgr A^∗ (a source centered on the BH) even closer than S2 have been announced by Peißker et al. (2020a, b).

S-stars have been extensively studied using spectroscopy (to obtain their radial velocities) and astrometry to understand their motion in three dimensions. Recently, Elaheh Hosseini et al. (2020) proposed to also use the photometry of S2 as a probe of the gaseous environment of Sgr A^∗. They suggested that the photometric signal of S2 may be perturbed by the bow shock that the star drives in the gaseous environment of Sgr A^∗, via either thermal or synchrotron emission at the shock. Although Elaheh Hosseini et al. (2020) found no evidence for the photometric variability of S2 at the level of $2.5\%$ (setting indirect constraints on the gas density near its pericenter), they have added stellar photometry to the toolbox of methods used to study the S-stars.

Due to the very high velocities of S-stars, in some cases reaching $\sim 0.1c$ at the pericenter, their brightness should be significantly modified by the phenomenon of Doppler boosting (also known as Doppler or relativistic beaming) — the process by which the relativistic effects change the apparent brightness of the moving object. This effect has been previously invoked for a variety of astrophysical objects, e.g. relativistic jets (Begelman et al., 1984), compact binaries (Shakura & Postnov, 1987; van Kerkwijk et al., 2010), stars hosting short-period exoplanets (Loeb & Gaudi, 2003; Mazeh & Faigler, 2010), and supermassive black hole binaries with accretion disks D’Orazio et al. (2015). Here, prompted in part by the recently announced new S-stars closely approaching the Galactic Center black hole (Peißker et al., 2020a, b), we demonstrate the importance of the Doppler boosting for some of these objects¹¹1Potential role of the Doppler boosting in the Galactic Center was briefly mentioned in Zucker et al. (2006), but then it was dismissed as hardly observable in Zucker et al. (2007). and show that the associated photometric signal can reach significant levels (several per cent).

As the star orbits the Galactic BH, its flux received by a distant observer changes due to both special and general relativistic effects — time dilation, aberration of light, change of the photon frequency due to the stellar motion and gravitational redshift. The combination of these effects results in Doppler boosting of the stellar emission.

We will be interested in the stellar flux received by the observer at rest far from the Galactic Center (corrections due to the Earth motion and non-zero gravitational potential of the observer can be trivially added, Blanchet et al. 2001) in a narrow filter with the window function $w(\nu)$ and central frequency $\nu_{\rm obs}$. If $I_{\nu,{\rm obs}}$ is the radiation intensity in the observer’s frame, then the flux in that filter measured by the observer is

Photon number conservation dictates that $I_{\nu}(\nu)/\nu^{3}$ is a relativistic invariant (Rybicki & Lightman, 1986), meaning that $I_{\nu,{\rm obs}}(\nu_{\rm obs})$ is related to the intensity in the stellar (emitter) frame $I_{\nu,{\rm em}}(\nu_{\rm em})$, at the frequency of emission $\nu_{\rm em}$ related to $\nu_{\rm obs}$ through the Doppler shift, as $I_{\nu,{\rm obs}}(\nu_{\rm obs})=I_{\nu,{\rm em}}(\nu_{\rm em})(\nu_{\rm obs}/\nu_{\rm em})^{3}$. Assuming that the filter is narrow and that $\nu_{\rm obs}/\nu_{\rm em}$ only slightly deviates from unity, we can also write

where

Plugging these relations into equation (1) and using the fact that $\nu_{\rm obs}/\nu_{\rm em}$ is independent of $\nu_{\rm obs}$ (see Eq. 5 below), one obtains

where the emitted flux, assumed to not vary in time, is $F_{\rm em}=\int_{0}^{\infty}I_{\nu,{\rm em}}(\nu)w(\nu){\rm d}\nu$.

To relate $\nu_{\rm em}$ to $\nu_{\rm obs}$ we use the results of Blanchet et al. (2001), which account for both special and general relativistic effects and are accurate up to $O(v^{2}/c^{2})$:

where $v_{\rm LOS}$ is the stellar line-of-sight velocity (assumed positive for receding objects) and $v$ is the full velocity of the emitting star, while $\Phi_{\rm BH}(r)=-GM_{\rm BH}/r$ is the BH potential at the instantaneous stellar distance $r$ ($M_{\rm BH}\approx 3.98\times 10^{6}$ M_⊙ is the BH mass, Do et al. 2019a). In this expression the terms in the denominator multiplied by $c^{-2}$ represent the $O(v^{2}/c^{2})$ correction due to the general relativistic redshift and the special relativistic transverse Doppler shift, respectively. They go beyond the standard $O(v/c)$ non-relativistic Doppler shift expression appearing in the numerator, and we explore if they can provide a measurable contribution to the stellar photometric signal in §3.

Equations (4) and (5) yield the following expression for the relative flux variation due to the Doppler boosting

which is valid to $O(v^{2}/c^{2})$ order. Retaining the terms only to first order in $v/c\ll 1$ one recovers the familiar result (Loeb & Gaudi, 2003; D’Orazio et al., 2015)

for the flux variation $\left(\Delta F/F_{\rm em}\right)_{1}$ accurate to $O(v/c)$.

Because of heavy dust obscuration, the S-stars are observed in the infrared. At these wavelength the emission of hot S-stars lies in the Rayleigh-Jeans tail of the blackbody spectrum, and we can safely assume $\alpha\approx 2$.

To compute flux variation as a function of time $t$ one needs to know how the stellar velocity $\mathbf{v}$ and distance $r$ relative to the BH vary with $t$. We assume ${\bf v}(t)$ and $r(t)$ to be given by their Keplerian values, neglecting the relativistic corrections (which provide a relative contribution at $O(v^{2}/c^{2})$ order, negligible at our level of accuracy), e.g. the GR pericenter precession²²2The largest pericenter shift for the stars considered in this study does not exceed $2^{\circ}$ per orbit (Peißker et al., 2020b)..

As in Do et al. (2019a), we account for the Roemer delay — the propagation time across the binary orbit (neglecting the Shapiro delay). The corresponding time correction is rather small (typically several days) as it is $\sim\langle v\rangle/c$ of the orbital period $P_{\rm orb}$, where $\langle v\rangle$ is the mean stellar velocity (which is considerably lower than the pericenter velocity for highly eccentric orbits). This time shift does not affect the amplitude of the Doppler boosting signal, which is of most interest for us.

We now describe the results of our calculation of the Doppler boosting signal for several S-stars in the Galactic Center, some of which have been recently discovered by Peißker et al. (2020a, b). We choose four objects³³3Note that GRAVITY Collaboration et al. (2020) challenged the determination of the orbital parameters of S62 by Peißker et al. (2020a, b), which may be considered as tentative. most promising from the point of view of revealing observable photometric variation, including S2, with the properties listed in Table 1. We illustrate their lightcurves (relative flux deviations in a narrow filter) computed using equation (6) in Figures 1-4 (middle panels), in which we also plot their line-of-sight velocity $v_{\rm LOS}(t)$ (top panels).

In the bottom panels of these figures we plot $(\Delta F/F_{\rm em})_{2}$, the photometric deviation due to $O(v^{2}/c^{2})$ effects — gravitational redshift and transverse Doppler shift — defined as $(\Delta F/F_{\rm em})_{2}=(\Delta F/F_{\rm em})-(\Delta F/F_{\rm em})_{1}$. Using equations (6) and (7) one can show for $\alpha=2$ that, to leading order,

where $v_{\perp}$ is the transverse velocity (in the sky plane). The first two terms in the right hand side represent specific total energy of the star, which is negative for bound objects, implying that $(\Delta F/F_{\rm em})_{2}<0$. Note that $(\Delta F/F_{\rm em})_{2}$ represents the $O(v/c)$ relative correction to the leading-order result (7), which is an $O(v/c)$ effects itself.

We now discuss our results for each individual star listed in Table 1 (its last two columns also list the ranges of variation of $\Delta F/F_{\rm em}$ and $(\Delta F/F_{\rm em})_{2}$).

S4711    This object has the shortest orbital period around the Galactic Center BH as claimed by Peißker et al. (2020b) — about 7.7 yr, see Table 1. However, the relatively low eccentricity places its pericenter at $r_{p}=142$ au, far enough from the BH to make $v_{\rm LOS}$ to change by only about $6800$ km s^-1 in the course of about 10 months. Over that period the photometric signal changes by $\approx 2.3\%$, while the $O(v^{2}/c^{2})$ effects contribute to this variation at the level of $0.04\%$.

S4714    This star has the third shortest orbital period among the currently claimed S-stars (see Table 1), but its very high eccentricity brings its pericenter to $r_{p}\approx 12.6$ au (although with large uncertainty, $\pm 9.3$ au, Peißker et al. 2020b). This results in large variation of the photometric signal at the level of $6.5\%$, of which $0.25\%$ comes from $O(v^{2}/c^{2})$ effects.

The orbit of S4714 has a rather peculiar orientation, with its apsidal axis lying almost in the plane of the sky, which explains the spiky behavior of $v_{\rm LOS}(t)$ and $\Delta F/F_{\rm em}$ in Figure 4. For that reason, to characterize the timescale of photometric variability we do not use the time between the $v_{\rm LOS}$ extrema, which is about half of $P_{\rm orb}$. Instead, we use the time it takes $|\Delta F/F_{\rm em}|$ of S4714 to stay above its half-maximum value, which is very short, about 5.3 d.

Unfortunately, due to its faintness (m${}_{K}=17.7$, Peißker et al. 2020b) this star may present less attractive target for the photometric followup than S62.

Calculations presented in Section 3 clearly demonstrate that Doppler boosting can potentially provide an interesting purely photometric probe of the orbital dynamics in the Galactic Center. Another source of photometric variability of the S-stars could be the changing amplitude and orientation of their tidal bulge induced by the black hole gravity. The amplitude of this effect at the pericenter distance $r_{p}$ can be estimated as (Morris, 1985)

where $M_{\star}$ and $R_{\star}$ are the stellar mass and radius. One can see that this effect is generally negligible compared to the Doppler boosting (its signal also drops faster with the distance from the BH).

We explored the importance of Doopler boosting for the photometric observations of S-stars in the Galactic Center. Focusing on four objects with very tight orbits around the BH, we demonstrated that Doppler boosting leads to photometric variations of these S-stars at the level of several per cent (around $6\%$ for S62 and S4714) around the time of pericenter passage. Measurement of this signal can be used for confirming and refining orbital parameters of the S-stars for which spectroscopy and astrometry are difficult. Doppler boosting should also be accounted for when S-stars are used as probes of the gaseous environment of the Sgr A^∗. Intrinsic variability of Sgr A^∗ and crowding of the surrounding stellar field are the biggest observational challenges that a successful measurement of Doppler boosting would need to overcome. Measuring higher order, $O(v^{2}/c^{2})$, effects due to transverse Doppler shift and GR redshift is unlikely to be be possible unless the systematic effects discussed in Section 4.2 are suppressed by orders of magnitude.