⚛️ general relativity

Effects of a disk structure on stellar motion at the Galactic Center

This paper analytically demonstrates that a flattened disk-like mass distribution at the Galactic Center induces unique secular orbital shifts and precessions in stars like S2 and S301, which can be distinguished from spherical mass effects and used to constrain the disk's properties and the black hole's spin.

Original authors: Arianna Foschi, Frederic H. Vincent, Thibaut Paumard, Guy Perrin

Published 2026-07-30

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Arianna Foschi, Frederic H. Vincent, Thibaut Paumard, Guy Perrin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the center of our galaxy, the Milky Way, as a cosmic dance floor. In the very middle sits the DJ, a supermassive black hole named Sagittarius A*, so heavy it crushes the space around it. Spinning around this DJ are the stars, the dancers, tracing elegant loops in the dark. For decades, astronomers have watched these dancers to learn secrets about the DJ, like how fast it spins. But to do this, they had to make a big assumption: they thought the crowd of other stars and gas surrounding the DJ was spread out perfectly evenly in a giant, round ball, like a fluffy cloud of cotton candy.

However, nature rarely does things perfectly round. In reality, the "crowd" at the galactic center looks more like a flattened pancake or a spinning vinyl record. This paper asks a simple but tricky question: What happens to the stars' dance moves if the crowd is a flat disk instead of a round ball? The authors use the star S2 (a famous, fast-moving dancer) and a newly discovered one called S301 to test this. They want to know if the shape of the crowd changes the dance steps enough to confuse our measurements of the black hole's spin. If we don't account for the flat disk, we might think the black hole is spinning when it's actually just the crowd pushing the star, or vice versa.

The Flat Disk vs. The Round Ball

The researchers, A. Foschi and colleagues, set out to simulate what happens when a star orbits a black hole that is surrounded by a thin, flat disk of matter, rather than a spherical cloud. They focused on two specific stars: S2, which we know very well, and S301, a new discovery that is expected to be very sensitive to the black hole's spin.

In their simulation, they treated the disk as a flat sheet of matter with a density that changes as you get closer to or further from the center, following a specific mathematical rule (a power law). They calculated the tiny gravitational tugs this disk would give to the stars as they zoomed around.

The Three New Moves

The team found that a flat disk doesn't just push the star in the same way a round ball of mass would. It introduces three specific "moves" or changes to the star's orbit that a round ball simply doesn't create:

The Stretch: The disk causes a slow, steady change in the shape of the star's orbit, specifically stretching or shrinking a value called the "semi-latus rectum" (a measure of the orbit's width). This is a new effect; neither the black hole's own gravity (at the level of precision they used) nor a round cloud of mass causes this. It's like the disk is gently pulling the dancer's arms in or out as they spin.
The Twist (In-Plane): The disk makes the star's closest approach to the black hole (the pericenter) shift slightly more or less than expected. While a round cloud does this too, the disk can do it in a way that mimics a cloud with ten times more mass. This means if we assume the mass is round, we might underestimate how much "stuff" is actually there if it's actually a disk.
The Tilt (Out-of-Plane): This is the most exciting find. The disk pushes the star up and down, out of its flat orbital plane. This "out-of-plane precession" is a wobble that a round cloud cannot create.

S2: The Perfect Detective

For the star S2, the authors found that the black hole's own spin (which causes a similar "tilt" effect called Lense-Thirring precession) is so weak that it's basically invisible. However, the tilt caused by the disk is strong enough to be seen with current telescopes.

The paper suggests that if we measure this specific "tilt" in S2's orbit, we can set a strict limit on how much mass is in the disk. The authors calculated that for a reasonable disk mass, the shift in the orbit's width is about 10 microarcseconds (a tiny angle, but measurable), and the tilt is about 0.03 degrees. Because the black hole's spin doesn't mess with S2's orbit in this way, S2 acts as a perfect detective: any tilt we see must be from the disk, not the black hole. This allows astronomers to weigh the disk without the black hole's spin confusing the results.

S301: The Spin Detective with a Twist

The star S301 is different. It orbits much closer to the black hole, making it sensitive to the black hole's spin. The authors found that the disk's "tilt" effect on S301 is about ten times smaller than the maximum possible spin effect.

However, there's a catch. The spin effect depends heavily on the angle of the black hole's spin axis, which we don't know yet. If the spin axis is tilted in a certain way, the spin effect could be tiny, making the disk's effect the dominant one. If the spin is aligned differently, the spin effect could be huge. This creates a "degeneracy," meaning the two effects can look similar and be hard to tell apart.

The paper argues that by first using S2 to figure out exactly how massive the disk is (since S2 isn't confused by the spin), we can then subtract the disk's effect from S301's data. This would leave us with a clean measurement of the black hole's spin, free from the confusion of the disk's gravity.

Ruling Out the "Big Neighbors"

Finally, the team checked if other giant structures nearby, like a massive cloud of gas called the Circumnuclear Disk or a giant molecular cloud named Sagittarius B2, could mess up these measurements. They simulated the gravitational pull of these massive neighbors and found that their effect is roughly 100 times weaker than the effect of the stellar disk right next to the black hole. So, we can safely ignore them; they are too far away to change the dance steps of S2 or S301.

The Bottom Line

This paper doesn't claim to have found a new black hole or measured the spin yet. Instead, it provides the mathematical toolkit to ensure that when we do measure the spin of Sagittarius A* using stars like S301, we aren't being tricked by a flat disk of stars.

The key takeaway is that the shape of the matter around the black hole matters. A flat disk leaves a unique fingerprint on the stars' orbits—specifically a steady stretch and a specific kind of tilt—that a round ball of mass cannot mimic. By understanding these fingerprints, astronomers can separate the "noise" of the disk from the "signal" of the black hole's spin, leading to a much clearer picture of the center of our galaxy.

Technical Summary: Effects of a disk structure on stellar motion at the Galactic Center

Problem Statement
Current models of stellar dynamics at the Galactic Center (GC) typically assume that the extended mass distribution surrounding the supermassive black hole (SMBH), Sagittarius A* (Sgr A*), is spherically symmetric (e.g., Plummer or cusp profiles). This assumption simplifies the disentanglement of the SMBH's relativistic effects from the gravitational influence of extended mass. However, observational evidence indicates the presence of a coherent, disk-like structure of young stars (the clockwise or CW disk) and theoretical arguments suggest that accretion flows and stellar remnants may also form flattened, disk-like distributions. The spherical assumption may therefore be insufficient for accurately modeling stellar orbits, particularly for the newly discovered star S301, which is a prime candidate for measuring the Sgr A* spin via Lense-Thirring (LT) precession. This paper investigates how a thin, axisymmetric disk structure affects the orbital elements of stars S2 and S301, specifically examining whether disk-induced perturbations can mimic or obscure relativistic spin effects.

Methodology
The authors develop a purely Newtonian framework to derive the acceleration exerted by a thin disk with a power-law surface density $\Sigma \propto r^{-\gamma}$ , where $\gamma$ ranges between 1 and 2.

Acceleration Derivation: Starting with the potential of a single uniform ring, the authors analytically integrate the gravitational force over a disk surface to obtain the acceleration components ( $a_d$ and $a_z$ ) in the disk frame. They account for the singularity at the disk plane ( $z=0$ ) and derive asymptotic limits for both inner ( $r \ll R$ ) and outer ( $r \gg R$ ) regimes.
Reference Frames: The study utilizes three reference frames: the BH/Disk frame, the orbital frame, and the observer (sky) frame. The transformation between these frames allows the disk acceleration to be projected into the Gaussian frame (radial, transverse, and normal components: $R, S, W$ ) co-moving with the star.
Osculating Formalism: Using the Gauss planetary equations, the authors compute the secular variations of orbital elements (semi-latus rectum $p$ , eccentricity $e$ , argument of pericenter $\omega$ , inclination $i$ , and longitude of ascending node $\Omega$ ) over one orbit. They incorporate first-order Post-Newtonian (1PN) terms for Schwarzschild precession and 1.5PN terms for LT precession to compare disk effects with relativistic effects.
Specific Configurations: The analysis focuses on two scenarios:
- A hypothetical general power-law disk with mass $M_{tot} = 10^3 M_\odot$ to maximize effects and explore parameter space.
- A realistic configuration based on the observed CW stellar disk ( $\gamma = 2$ ), incorporating luminous mass ( $M_L \sim 10^4 M_\odot$ ) and a hypothetical non-luminous mass component ( $M_{extra}$ ) extending inward.

Key Contributions and Results
The paper derives analytical expressions for the secular shifts induced by a disk and applies them to S2 and S301. The key findings are:

Secular Shift in Semi-Latus Rectum ( $\Delta p_{star}$ ): Unlike spherical distributions or low-order PN terms, a disk structure induces a secular variation in the semi-latus rectum. For S2, with a disk mass of $10^3 M_\odot$ , this shift is $\sim 10 \, \mu\text{as}$ , which is commensurate with current astrometric precision. This effect serves as a distinctive signature of non-spherical extended mass.
In-Plane Precession ( $\Delta \omega_{obs}$ ): The disk induces an in-plane precession comparable to that of a Plummer distribution with similar mass. However, for specific orientations and higher masses (e.g., $10^4 M_\odot$ ), the disk can produce secular trends similar to lower-mass spherical models, suggesting that disk structures can accommodate more mass than currently constrained by spherical limits without violating observational data.
Out-of-Plane Precession ( $\Delta \Theta$ ): The disk generates a normal component of acceleration ( $W_{disk}$ $W_{d i s k}$ ) leading to out-of-plane precession.
- For S2: The LT precession is negligible. The disk-induced out-of-plane precession ( $\Delta \Theta \sim 0.03^\circ$ ) is comparable to current measurement precision. This allows S2 to be used to place upper limits on the non-luminous mass in disk-like structures, independent of spin degeneracies.
- For S301: The disk effect is generally smaller than the maximum LT precession (by a factor of $\sim 10$ ) assuming a maximally spinning black hole with a face-on orbit. However, if the spin axis is inclined or the spin magnitude is lower, the disk precession can become substantial, potentially dominating the signal.
CW Disk Application: Applying the model to the specific geometry of the CW disk ( $\gamma=2$ ), the authors find that S2 data can constrain the non-luminous mass within the disk. They show that a disk mass up to $\sim 10^6 M_\odot$ (100 times the luminous mass) could be compatible with S2's motion if the mass is concentrated inward.
Degeneracy Breaking: The paper demonstrates that the disk and spin effects behave differently with respect to inclination angles. While the spin inclination $\theta$ affects in-plane and out-of-plane precessions in opposite ways, varying the disk inclination $i_{BH}$ shifts both in the same direction. This difference offers a pathway to disentangle the two effects.
External Structures: The study confirms that external structures like the Circumnuclear Disk (CND) and the giant molecular cloud Sagittarius B2 have negligible effects on S2 and S301 orbits compared to the inner CW disk.

Significance
The authors conclude that assuming a spherical mass distribution may lead to incorrect constraints on the extended mass and the spin of Sgr A*. The presence of a disk structure introduces unique secular signatures (specifically in $p$ and out-of-plane precession) that are absent in spherical models.

The primary significance lies in the ability to use S2 to constrain the mass of disk-like structures without the degeneracy of the Sgr A* spin, which is currently negligible for S2. These constraints are crucial for interpreting future measurements of S301. If the disk mass is not independently constrained, the disk-induced precession could be misinterpreted as the LT precession (or vice versa), leading to biased measurements of the black hole's spin. The paper emphasizes that future monitoring of stellar orbits must account for extended disk structures to accurately characterize the central compact object. The results are reproducible using the open-source Python package PERSEO.