The effects of the spin and quadrupole moment of SgrA* on the orbits of S stars

This paper analytically and numerically characterizes the 2PN-order orbital precessions of S-stars induced by the spin and quadrupole moment of Sgr A*, providing theoretical expressions and observer-independent insights to facilitate future constraints on the black hole's properties using GRAVITY+ observations.

The Gist

Imagine the center of our galaxy, the Milky Way, as a cosmic dance floor. At the very center spins a massive, invisible partner: a supermassive black hole named Sgr A*. Around it, a group of stars (called "S-stars") are performing a high-speed waltz.

For years, astronomers have watched these stars to understand the rules of gravity. But this new paper is like a detective looking for a very specific, subtle clue in the dance: how the black hole's spin and shape affect the stars' paths.

Here is the breakdown of the paper's story, explained simply.

1. The "Hairless" Mystery

In physics, there's a famous rule called the "No-Hair Theorem." It says that a black hole is incredibly simple. No matter how complex it was when it formed, once it settles down, it only has three "features" (or "hairs") that we can measure from the outside:

1. Mass (How heavy it is).
2. Spin (How fast it's rotating).
3. Charge (How much electricity it holds).

For Sgr A*, we know its mass very well. We know its charge is basically zero. But we don't know its spin or its exact shape yet. If we can measure the spin and the shape independently, and they match the "No-Hair" prediction, we prove Einstein's theory of General Relativity is perfect. If they don't match, we might need new physics!

2. The Dance Moves: Three Types of Wobbles

When a star orbits a spinning black hole, its path doesn't just stay in a perfect, flat circle. It wobbles. The paper identifies three main types of wobbles:

• The "Schwarzschild" Wobble (The Big One): This is the most obvious effect. The orbit isn't a closed loop; it's like a flower petal that slowly rotates forward every time the star goes around. We've already seen this with the star S2.
• The "Lense-Thirring" Wobble (The Frame-Dragging): Imagine the black hole is a giant blender spinning in a thick smoothie. As it spins, it drags the smoothie (space itself) along with it. If a star orbits near it, the black hole's spin tries to twist the star's orbit sideways. This is like the blender pulling the star's dance floor into a spiral.
• The "Quadrupole" Wobble (The Shape Shift): A spinning black hole isn't a perfect sphere; it's slightly squashed at the poles and bulging at the equator (like a spinning pizza dough). This "squash" creates a gravitational tug that also twists the star's orbit, but in a different way than the spin drag.

The Problem: The "Big Wobble" (Schwarzschild) is huge. The "Spin" and "Shape" wobbles are tiny—like trying to hear a whisper in a hurricane. For the star S2, which is about 120 AU away from the black hole, these tiny whispers are almost impossible to hear.

3. The Solution: Find a Closer Dancer

The authors realized that to hear the "whispers" of the spin and shape, we need a star that dances much closer to the black hole.

They proposed a hypothetical star they call "S2/10."

• S2 is the current star we watch.
• S2/10 is a star with the same shape of orbit but 10 times closer to the black hole.

The Analogy: Imagine you are trying to hear a pin drop.

• If you stand 10 meters away (S2), you can't hear it over the wind.
• If you stand 1 meter away (S2/10), the pin drop sounds like a gunshot.

Because gravity gets stronger the closer you are, the effects of the black hole's spin and shape become massively amplified for S2/10.

• The "Spin" effect becomes 1,000 times stronger.
• The "Shape" effect becomes 3,000 times stronger.

4. The New Telescope: GRAVITY+

The paper mentions a new upgrade to our telescopes called GRAVITY+. Think of this as upgrading from a pair of binoculars to a high-powered microscope. This new instrument will be sensitive enough to spot these faint, close-in stars (like S2/10) that are currently too dim to see.

5. What the Paper Did

The authors did two main things:

1. The Math: They wrote down complex equations (using something called "Post-Newtonian" math) to predict exactly how these close stars should move if Einstein's theory is right. They calculated how the orbit should twist, tilt, and rotate over time.
2. The Simulation: They built a computer code (named OOGRE) to simulate these stars. They ran the numbers and confirmed that their math matches the computer simulation.

6. The Big Takeaway

The paper concludes that to test the "No-Hair Theorem" and understand the black hole's spin, we don't just need better math; we need closer stars.

• If we find S2/10: We can measure the black hole's spin and shape with incredible precision.
• If we can't find S2/10: We can still try to combine data from many different stars, but it will take much longer.

In a nutshell: This paper is a roadmap for the future. It tells astronomers, "Don't just watch the stars far away; look deeper, look closer, and you will finally see the black hole's spin and shape, proving whether the universe follows the rules Einstein wrote 100 years ago."

Technical Summary

1. Problem Statement and Context

The paper addresses the challenge of testing the "no-hair" theorem for the supermassive black hole (SMBH) Sgr A* at the center of the Milky Way. According to this theorem, a black hole is fully characterized by only three parameters: mass ($M$), spin (angular momentum, $J$), and electric charge. Since the charge is negligible, the theorem implies a specific relationship between the mass and the quadrupole moment ($Q$) of the gravitational field: $Q = -J^2/M$.

While the Schwarzschild precession (due to mass) has been detected in the orbit of the star S2, detecting the higher-order Kerr effects—specifically the Lense-Thirring (LT) effect (frame-dragging due to spin) and the quadrupole moment effect—remains elusive. These effects are significantly weaker and decay rapidly with distance from the black hole. The authors aim to analytically and numerically characterize these effects to determine if future observations (e.g., with GRAVITY+) can constrain the spin and quadrupole moment, potentially using hypothetical stars closer to Sgr A* than S2.

2. Methodology

Theoretical Framework

• Post-Newtonian (PN) Approximation: The authors derive analytical expressions for the secular evolution of orbital parameters up to the 2nd Post-Newtonian (2PN) order.
  • Schwarzschild effect: Dominant at 1PN, with subleading terms at 2PN.
  • Lense-Thirring effect: Dominant at 1.5PN (order $(v/c)^3$).
  • Quadrupole moment effect: Dominant at 2PN (order $(v/c)^4$).
• Reference Frames: The study utilizes four reference frames:
  1. Fundamental Frame: Chosen as the observer's sky plane (Right Ascension/Declination).
  2. Orbit Frame: Defined by the instantaneous orbital plane and pericenter.
  3. Black Hole Frame: Aligned with the black hole's spin axis.
  4. Gaussian Frame: Radial and transverse vectors attached to the star.
• Analytical Derivation: Using the Lagrange planetary equations and the two-timescale method, the authors derive secular shifts for orbital elements (semi-latus rectum $p$, eccentricity $e$, inclination $\iota$, node $\Omega$, and argument of periapsis $\omega$).
• Observer-Independent Quantities: To avoid coordinate singularities (e.g., when inclination $\iota = 0$), the authors introduce observer-independent precession rates:
  • $\dot{\varpi}$: In-plane precession rate.
  • $\dot{\Theta}$ and $\dot{\Xi}$: Out-of-plane precession rates (rotation of the major and minor axes, respectively).

Numerical Simulation

• OOGRE Code: The authors utilize and extend their Python-based code, OOGRE (Osculating Orbits in General Relativistic Environments).
• Extensions: The code was upgraded from 1PN to include 2PN Schwarzschild terms, 1.5PN Lense-Thirring terms, and 2PN quadrupole moment terms.
• Simulation Target: A hypothetical star named "S2/10" is simulated. It shares S2's orbital parameters (eccentricity, etc.) but has a semi-major axis 10 times smaller (bringing it much closer to the event horizon), thereby amplifying relativistic effects.

3. Key Contributions

1. Derivation of 2PN Secular Shifts: The paper provides explicit analytical formulas for the secular shifts of the pericenter ($\Delta \varpi$), the major axis tilt ($\Delta \Theta$), and the minor axis tilt ($\Delta \Xi$) up to 2PN order.
2. Decomposition of Precession: The authors distinguish between:
   • In-plane precession: Rotation of the pericenter within the orbital plane.
   • Out-of-plane precession: Reorientation of the orbital plane itself (nodal precession).
3. Observer Independence: By defining precession rates based on the orbit frame rather than the observer's sky plane, the authors provide robust quantities ($\varpi, \Theta, \Xi$) that remain valid even for face-on or edge-on orbits where standard Euler angles become degenerate.
4. Scaling Laws for Closer Orbits: The study quantifies how these effects scale for stars closer to the black hole. For a star with a semi-major axis 10 times smaller than S2 ("S2/10"):
   • Schwarzschild precession increases by $\sim 300\times$ (over one S2 period).
   • Lense-Thirring precession increases by $\sim 1000\times$.
   • Quadrupole moment precession increases by $\sim 3000\times$.

4. Key Results

Analytical Findings

• In-Plane Precession ($\Delta \varpi$):
  • Schwarzschild: Always positive (prograde), independent of spin orientation.
  • Lense-Thirring: Depends on the angle $\theta$ between the spin and orbital angular momentum. It is maximal for equatorial orbits ($\theta=0$) and vanishes for polar orbits. It causes a retrograde shift for prograde orbits and a prograde shift for retrograde orbits relative to the spin.
  • Quadrupole: Depends on $\cos^2\theta$. It is maximal for both equatorial and polar orbits and always adds to the Schwarzschild precession (prograde).
• Out-of-Plane Precession ($\Delta \Theta, \Delta \Xi$):
  • Schwarzschild: Contributes zero to out-of-plane precession.
  • Lense-Thirring: Maximal for polar orbits ($\theta = \pi/2$). It causes the orbital plane to precess around the black hole's spin axis.
  • Quadrupole: Maximal for orbits with an inclination of $\theta = \arccos(\pm 1/\sqrt{3}) \approx 54.7^\circ$. It induces a precession of the orbital plane that depends on the orientation of the spin relative to the orbit's axes.

Numerical Simulations (S2 vs. S2/10)

• S2: The relativistic shifts are detectable but small. The quadrupole moment effect is currently below the detection threshold of current instruments.
• S2/10: The simulations confirm that for a star 10 times closer, the accumulated angular shifts over a single S2 period are massive.
  • The Lense-Thirring effect dominates the out-of-plane reorientation for polar orbits.
  • The Quadrupole effect significantly alters the in-plane pericenter shift and out-of-plane tilt for specific inclinations.
• Configuration Dependence: The paper demonstrates that no single orbital configuration maximizes all effects simultaneously.
  • Equatorial orbits maximize in-plane precession but have zero out-of-plane LT precession.
  • Polar orbits maximize out-of-plane LT precession but have zero in-plane LT precession.
  • Mid-inclined orbits are required to maximize the quadrupole out-of-plane effect.

5. Significance and Conclusion

• Testing the No-Hair Theorem: The paper establishes that measuring the spin and quadrupole moment independently is necessary to verify the relation $Q = -J^2/M$. Current data on S2 is insufficient; the detection of these higher-order effects requires stars significantly closer to Sgr A*.
• Role of GRAVITY+: The upcoming GRAVITY+ instrument, with its improved sensitivity to fainter stars, is identified as the critical tool for detecting these "S2/10" candidates.
• Multi-Star Strategy: Even if new close-in stars are not found, the paper suggests that multi-star fitting using the diverse orientations of currently known S-stars can provide the necessary constraints to disentangle spin and quadrupole effects.
• Scientific Impact: By providing observer-independent analytical expressions and validating them with numerical codes, the authors offer a robust framework for interpreting future high-precision astrometric data. This will allow astronomers to probe the multipolar structure of Sgr A* and potentially confirm the Kerr metric nature of the Galactic Center black hole.

In summary, this work bridges the gap between theoretical General Relativity and future observational capabilities, demonstrating that the detection of the black hole's spin and quadrupole moment is feasible but requires targeting stars in the immediate vicinity of the event horizon.