Detection of simultaneous QPO triplets in 4U 1728-34 and constraining the neutron star mass and moment of inertia

By detecting simultaneous twin kHz and $\sim 40$ Hz QPO triplets in 4U 1728-34 and interpreting them as orbital, periastron, and nodal precession frequencies within a Kerr metric framework, the study constrains the neutron star's mass and moment of inertia to favor stiffer equations of state.

The Gist

Imagine a cosmic dance floor where a tiny, incredibly dense star (a neutron star) is spinning so fast it's a blur, and a cloud of hot gas is swirling around it like water down a drain. This is the setting of the paper you're asking about. The scientists are trying to figure out exactly how heavy this star is and how "stiff" its insides are, just by listening to the music it makes.

Here is the story of their discovery, broken down into simple concepts:

1. The Cosmic Music (Quasi-Periodic Oscillations)

When matter falls onto a neutron star, it doesn't just fall silently. It vibrates and pulses, creating rhythmic "beats" in X-ray light. Astronomers call these Quasi-Periodic Oscillations (QPOs).

Think of these QPOs like the notes on a piano.

• Low notes: A slow, deep rumble (around 40 Hz).
• High notes: Two very fast, high-pitched trills (around 800 Hz and 1100 Hz).

Usually, astronomers see these notes one by one or in pairs. But in this study, using a powerful Indian space telescope called AstroSat, the team caught a rare moment where three distinct notes were playing at the exact same time. It was like hearing a perfect musical chord for the first time.

2. The Dance Moves (Relativistic Precession)

Why do these three specific notes happen together? The scientists used a theory called the Relativistic Precession Model (RPM).

Imagine a marble rolling around a giant, spinning bowling ball (the neutron star).

• The Orbit (The Upper Note): The marble goes around the ball in a circle. This is the fastest frequency.
• The Wobble (The Lower Note): Because the ball is so heavy, the marble's path isn't a perfect circle; it's an oval that slowly rotates (precesses) like a spinning top. This creates a second, slightly slower beat.
• The Tilt (The Low Note): The marble also wobbles up and down, like a coin spinning on a table. This creates the slowest beat.

The paper found that the three notes they heard matched the math of these three specific dance moves perfectly.

3. Weighing the Invisible Star

Once they knew the "dance moves" matched the theory, they could use the speed of the music to weigh the star.

• The Problem: Neutron stars are so dense that a teaspoon of their material would weigh a billion tons. We can't put them on a scale.
• The Solution: By measuring the exact frequency of the "notes," the scientists could calculate the star's Mass (how heavy it is) and its Moment of Inertia (how its mass is distributed inside—like whether it's a solid rock or a hollow shell).

The Result: They found the star weighs about 1.92 times the mass of our Sun. That's heavy!
They also calculated the "Moment of Inertia," which tells us about the star's internal structure.

4. The "Jelly" vs. "Rock" Debate (Equation of State)

This is the most exciting part. Neutron stars are made of matter so dense that our normal laws of physics break down. Scientists have different theories (called Equations of State) about what this stuff is like inside:

• Soft/Runny: Like jelly or honey.
• Stiff/Rocky: Like a super-hard diamond or a solid metal.

The "Moment of Inertia" acts like a fingerprint. If the star is "jelly-like," it spins and wobbbs one way. If it's "rock-hard," it spins differently.

The data from this paper suggests the neutron star is made of stiff material. It's more like a super-dense diamond than a blob of jelly. This helps rule out some theories about how matter behaves under extreme pressure.

5. The "Kerr" Approximation (The Map vs. The Territory)

The scientists used a mathematical map (called the Kerr metric) to do their calculations. This map is perfect for black holes, but neutron stars are slightly different because they have a physical surface and a specific internal structure.

The authors admit: "We used a map designed for black holes to navigate a neutron star."

• The Analogy: It's like using a map of the ocean to navigate a river. It's close, but not perfect.
• The Impact: They calculated that using this "black hole map" might make their weight estimate off by about 1% and their "stiffness" estimate off by about 5%.
• The Future: They plan to do a more precise calculation next time using a map specifically designed for neutron stars, which will give them an even sharper picture.

The Big Takeaway

This paper is a major step forward because:

1. It caught a rare triplet: Finding three synchronized QPOs at once is like catching a shooting star, a meteor, and a comet in the same second.
2. It weighs the unweighable: It gives us a very precise weight for a neutron star.
3. It reveals the interior: It tells us that the stuff inside these stars is incredibly "stiff," helping us understand the fundamental building blocks of the universe.

In short, by listening to the cosmic rhythm of a spinning star, these scientists have learned exactly how heavy it is and what its "insides" are made of, bringing us one step closer to understanding the most extreme matter in the universe.

Technical Summary

1. Problem Statement

High-frequency Quasi-Periodic Oscillations (HF QPOs) in Low-Mass X-ray Binaries (LMXBs) containing Neutron Stars (NS) are believed to originate from the innermost regions of the accretion flow, close to the compact object. While the Relativistic Precession Model (RPM) has been successfully applied to Black Hole (BH) systems to estimate mass and spin, its application to NS systems is complicated by the unknown Equation of State (EOS) of the neutron star matter.

• The Challenge: To constrain the NS mass ($M$) and Moment of Inertia ($I$), one requires the simultaneous detection of three QPO frequencies (a triplet) corresponding to the orbital frequency ($\nu_\phi$), periastron precession ($\nu_{per}$), and nodal precession ($\nu_{nod}$).
• The Gap: Previous observations of NS sources like 4U 1728-34 often lacked simultaneous detection of all three frequencies or showed complex, evolving features that were difficult to model with a single static triplet. Furthermore, the RPM assumes a Kerr metric (vacuum solution for BHs), whereas a spinning NS has a metric dependent on its internal structure (EOS), introducing systematic uncertainties.

2. Methodology

Data Acquisition and Reduction:

• Instrument: The study utilized data from the Large Area X-ray Proportional Counter (LAXPC) onboard the AstroSat satellite.
• Source: The neutron star LMXB 4U 1728-34.
• Observation: Data from March 7–8, 2016 (Observation ID: T01 041T01 9000000362), covering 20 orbits.
• Processing:
  • Thermonuclear bursts were removed from the light curve.
  • The remaining persistent emission (3–30 keV) was analyzed.
  • The observation was divided into 16 time segments to track the evolution of QPO frequencies, as the features were observed to vary significantly over time.

Modeling and Analysis:

• Spectral Fitting: Power Density Spectra (PDS) for each segment were fitted using six Lorentzian components and a power-law (zero index) to account for residual Poisson noise.
• Triplet Identification: The authors identified simultaneous detections of:
  1. Low-Frequency (LF) QPO ($\sim$35–45 Hz).
  2. Lower kHz QPO ($\sim$670–850 Hz).
  3. Upper kHz QPO ($\sim$1030–1140 Hz).
• Relativistic Precession Model (RPM) Application:
  • The observed frequencies were mapped to RPM frequencies: $\nu_{upper} = \nu_\phi$, $\nu_{lower} = \nu_{per}$, and $\nu_{LF} = \nu_{nod}$ (or $2\nu_{nod}$).
  • The authors tested the hypothesis that the LF QPO corresponds to twice the nodal precession frequency ($2\nu_{nod}$), a scenario suggested by the geometry of tilted inner accretion disks.
  • Metric Assumption: The analysis initially assumed a Kerr metric (standard for rotating black holes) to derive analytical expressions relating frequencies to Mass ($M$) and the ratio of Moment of Inertia to Mass ($I/M$).
  • Systematic Error Estimation: To account for the deviation of a real NS metric from the Kerr metric, the authors estimated the error by ignoring second-order spin terms ($\tilde{a}^2$) in the equations.

Equation of State (EOS) Constraints:

• The derived $M$ and $I$ values were compared against theoretical predictions.
• The Tolman-Oppenheimer-Volkoff (TOV) equations were solved numerically for 10 different EOSs (e.g., WFF2, SLY, AP4, ALF2) to generate $I$ vs. $M$ curves.
• Constraints from the gravitational wave event GW170817 (tidal deformability) were overlaid for comparison.

3. Key Contributions

1. First Detection of Multiple Triplets: The paper reports the first-ever detection of multiple sets of simultaneous QPO triplets (13 distinct triplets) from a single observation of a single NS source. This provides a robust statistical sample to test the RPM.
2. Refined Mass and Moment of Inertia Constraints: By identifying the LF QPO as $2\nu_{nod}$, the authors derived precise constraints on the NS parameters:
   • Mass: $M = 1.92 \pm 0.01 M_\odot$ (statistical) $\pm 0.03 M_\odot$ (systematic).
   • Moment of Inertia Ratio: $I_{45}/M = 1.07 \pm 0.01$ (statistical) $\pm 0.05$ (systematic), where $I_{45}$ is in units of $10^{45}$g cm$^2$.
3. Quantification of Metric Systematics: The study explicitly quantifies the systematic error introduced by using the Kerr metric for a NS. They found that the Kerr assumption leads to a $\sim$1% error in Mass and a $\sim$5% error in the Moment of Inertia ratio.
4. EOS Discrimination: The derived parameters were used to rule out softer EOSs, favoring stiffer equations of state (specifically WFF2, SLY, AP4, and ALF2).

4. Results

• QPO Correlations: The frequencies of the 13 triplets showed strong correlations consistent with the RPM predictions. The lower kHz QPO frequency increased from $\sim$674 Hz to $\sim$853 Hz, while the upper kHz QPO increased from $\sim$1036 Hz to $\sim$1136 Hz.
• Parameter Estimation:
  • Assuming the LF QPO is the fundamental nodal frequency ($\nu_{nod}$), the fit yielded a mass of $2.12 M_\odot$and an$I/M$ ratio of 2.21, which is theoretically inconsistent with standard NS models.
  • Assuming the LF QPO is twice the nodal frequency ($2\nu_{nod}$), the fit yielded$M = 1.92 M_\odot$and$I_{45}/M = 1.07$. This result is physically consistent with theoretical NS models.
• EOS Constraints: The derived $M$ and $I$ values fall within the parameter space favored by stiffer EOSs. The results are consistent with the constraints derived from the GW170817 gravitational wave event (specifically the 90% confidence interval for tidal deformability).
• Harmonic Nature: The authors discuss the possibility that the LF QPO is a harmonic. While the RMS of the signal ($\sim$5%) suggests it is the fundamental frequency, the necessity of identifying it as $2\nu_{nod}$ for the RPM to work implies a dynamic system geometry (e.g., a tilted disk) that enhances the second harmonic signal.

5. Significance

• Probing the Neutron Star Interior: This work demonstrates that simultaneous QPO triplets can be used to constrain the Moment of Inertia, a fundamental property that directly links to the density distribution and stiffness of matter inside the neutron star.
• Validation of RPM: The successful application of the RPM to a NS system with multiple triplets strengthens the model's validity, suggesting that the QPOs are indeed governed by relativistic orbital dynamics near the innermost stable circular orbit (ISCO).
• Future Directions: The study highlights that while the Kerr metric provides a good first-order approximation, future work must employ EOS-dependent metrics (computed numerically) to reduce systematic errors below the 5% level. This will allow for unprecedented constraints on the nuclear Equation of State, potentially distinguishing between different models of dense matter.
• AstroSat Capabilities: The results showcase the high timing resolution and sensitivity of the AstroSat/LAXPC instrument, capable of resolving complex, evolving QPO features in faint X-ray sources.