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Pulse profile modelling of the accretion-powered millisecond pulsar SAX J1808.4-3658 using NICER data from its 2019 and 2022 outbursts

This study applies pulse profile modelling to NICER data from the 2019 and 2022 outbursts of the accretion-powered millisecond pulsar SAX J1808.4-3658, revealing that while flexible background models improve fits, robust parameter constraints remain elusive due to degeneracies in non-pulsed radiation sources, highlighting the need for advanced accretion disc modelling and complementary high-energy polarimetric data.

Original authors: Bas Dorsman, Tuomo Salmi, Anna L. Watts, Mason Ng, Anna Bobrikova, Vladislav Loktev, Juri Poutanen, Joern Wilms

Published 2026-01-23

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CC BY 4.0

Original authors: Bas Dorsman, Tuomo Salmi, Anna L. Watts, Mason Ng, Anna Bobrikova, Vladislav Loktev, Juri Poutanen, Joern Wilms

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 a neutron star as a cosmic lighthouse. It spins incredibly fast, and as it rotates, beams of X-ray light sweep across space like the beam of a lighthouse. When these beams hit us on Earth, they look like a rhythmic pulse. By studying the shape and brightness of these pulses, scientists try to figure out the lighthouse's size, weight, and how it's tilted. This is the goal of "Pulse Profile Modelling."

This paper focuses on a specific lighthouse called SAX J1808.4−3658. Unlike most lighthouses that run on their own battery (rotation-powered), this one is powered by a "feast." It is a hungry star that is actively eating gas from a swirling disk of material (an accretion disk) orbiting around it. This makes it much brighter and more complex to study.

The researchers used data from NICER, a high-tech X-ray camera on the International Space Station, which took pictures of this star during two "feast" events (outbursts) in 2019 and 2022.

Here is the story of what they found, explained simply:

1. The "One Spot" Theory Didn't Work

At first, the scientists tried to explain the star's light using a simple model: imagine the star has just one hot, glowing spot on its surface (like a single campfire on a dark planet).

The Result: It was a bad fit. The model was like trying to fit a square peg in a round hole. The data showed extra "bumps" in the light that the single spot couldn't explain. Specifically, there was a strange glow at a specific energy level (around 1 keV) that looked like a reflection off the inner edge of the gas disk, which the simple model missed.

2. Adding a Second Spot Helped, But Not Enough

Next, they tried a model with two hot spots (two campfires). This is more realistic because the gas usually hits the star at two opposite magnetic poles.

The Result: This was better at explaining the rhythm of the pulses, but it still failed to explain the "glow" from the gas disk. The model was still missing a piece of the puzzle regarding how the gas disk shines.

3. The "Flexible Background" Trick

Since they couldn't perfectly model the gas disk's light, the researchers tried a clever workaround. Instead of trying to describe the gas disk with a specific physics formula, they let the computer freely adjust the background light to match whatever the data showed. Think of it like this: instead of trying to guess the exact color of a messy background painting, they just told the computer, "Make the background whatever color it needs to be to make the picture look right."

The Result: This worked beautifully! The math finally matched the data. The "messy" background was accounted for, and the pulse shapes fit perfectly.

4. The Big Problem: The "Who Did It?" Confusion

Here is the catch. Because the researchers let the background be so flexible, the computer got confused about who was producing the light.

The Dilemma: The computer couldn't tell if the steady, non-pulsing light was coming from the gas disk (the background) or from the star's hot spots (if the spots were huge and always visible).
The Analogy: Imagine you are in a room with a dim lightbulb and a flashlight. If you can't tell which one is on, you can't measure the flashlight's brightness accurately. In this case, the "flashlight" is the star's hot spots, and the "dim bulb" is the gas disk.
The Consequence: Because of this confusion, the scientists couldn't be sure about the star's mass and radius. Depending on how they set the rules for the background, the star could be calculated as either very heavy and small, or lighter and larger. The results were "degenerate," meaning multiple different answers could explain the same data.

5. What They Learned and What's Next

The paper concludes that while they made a huge step forward by using NICER data and a flexible background, they haven't solved the mystery of the star's exact size and weight yet.

The Missing Piece: They need better models for how the gas disk shines and how the hot spots look (maybe they aren't perfect circles?).
The Future: They suggest that adding polarization data (which measures the direction of the light waves, like sunglasses filtering glare) and data from higher energy levels will act like a second pair of eyes. This will help separate the "flashlight" from the "dim bulb," finally allowing them to measure the star's true mass and size with confidence.

In short: The scientists tried to weigh and measure a cosmic lighthouse by studying its light pulses. They found that their old maps (models) were too simple. While a new, flexible map worked better, it left them unsure if the light was coming from the lighthouse or the fog around it. To get a precise measurement, they need better tools to cut through that fog.

Technical Summary: Pulse Profile Modelling of SAX J1808.4−3658 using NICER Data

Problem Statement
Pulse Profile Modelling (PPM) is a relativistic ray-tracing technique successfully applied to rotation-powered millisecond pulsars (RMPs) to constrain neutron star (NS) mass and radius, thereby probing the dense matter equation of state (EoS). However, applying PPM to accretion-powered millisecond pulsars (AMPs) presents distinct challenges. Unlike RMPs, AMPs exhibit significant non-pulsed X-ray emission from the accretion disc and accretion stream, which complicates the separation of the pulsed hotspot signal from the background. Previous applications to AMPs, such as Salmi et al. (2018) using Rossi X-ray Timing Explorer (RXTE) data, relied on empirical models for Comptonization and archival data. This work addresses the need to apply modern, physically motivated PPM techniques to new Neutron star Interior Composition Explorer (NICER) data, specifically targeting the canonical AMP SAX J1808.4−3658 (J1808) during its 2019 and 2022 outbursts. The primary challenge lies in accurately modelling the accretion disc contribution and the complex geometry of the hotspots to avoid degeneracies that bias parameter inference.

Methodology
The authors employed Bayesian parameter inference using the X-PSI software package to fit pulse profiles from the 2019 and 2022 NICER outbursts of J1808. The analysis utilized a relativistic ray-tracing framework incorporating the oblate Schwarzschild approximation, gravitational redshift, light bending, and Doppler boosting.

Key methodological components included:

Hotspot Models: The study compared a single circular hotspot model (Single Temperature, ST) against a two-hotspot model (ST-U, where parameters are unshared). The hotspot atmospheres were modelled using a Compton slab atmosphere derived by Bobrikova et al. (2023), which assumes Compton scattering in a slab geometry of hot electrons.
Background/Disc Modelling: Three approaches were tested for the non-pulsed emission:
1. Disc Model: A multicolour disc blackbody (diskbb) model.
2. Disc + Line Model: The diskbb model augmented with a Gaussian line to account for broadened reflection features observed around 1 keV.
3. Marginalised Background: A flexible approach where the accretion disc model was replaced by a background component with counts marginalised over each energy bin independently, constrained by a fiducial flux and a multiplicative support factor ( $s$ ).
Priors: Priors were set for mass ( $M$ ), equatorial radius ( $R_{eq}$ ), distance, inclination, and hotspot geometry. In addition to wide, uninformative priors, the authors tested an EoS-informed prior based on the dense matter EoS constraints from Rutherford et al. (2024) to explore computational efficiency and parameter consistency.
Data Selection: Data were selected from the peak of the outbursts (August 2019 and August 2022) to ensure sufficient photon counts, with pulse profiles folded using updated timing solutions.

Key Results

Insufficiency of Single Hotspot Models: Single hotspot models (ST) failed to fit the data adequately across all background treatments. Significant residuals remained, particularly a spectral bump around 0.8–1.1 keV and phase-dependent residuals, indicating that a single circular hotspot cannot explain the observed pulse profiles.
Two-Hotspot Models with Disc Background: Introducing a second hotspot (ST-U) improved the phase residuals but failed to resolve the spectral residuals when using a physical disc model (diskbb or diskbb + Gaussian). The model still overestimated counts in certain bands and underestimated the narrow spectral feature near 1 keV. The inferred accretion rates were also significantly lower than independent estimates.
Success of Marginalised Background: The best statistical fits were achieved using the two-hotspot model (ST-U) combined with the marginalised background approach. This configuration yielded $\chi^2$ /DOF values consistent with a good fit (p-values of 0.17 and 0.30 for 2019 and 2022, respectively).
Parameter Degeneracy and Robustness: Despite the improved fit, the inferred parameters (mass, radius, inclination) were found to be non-robust due to a strong degeneracy between the non-pulsed background and the non-pulsed component of the hotspots.
- When the support factor ( $s$ ) for the marginalised background was increased from 100 to 1000, the inferred mass decreased from $\sim 2.0 M_\odot$ to $\sim 1.8 M_\odot$ , and the radius shifted from $\sim 8.7$ km to $\sim 8.1$ km.
- The choice of the support boundary significantly altered the posterior distributions, suggesting that without a physics-based upper limit on the background, the parameters are not uniquely constrained.
EoS-Informed Priors: Applying EoS-informed priors reduced computational cost by a factor of $\sim 2.5$ and shifted the inferred radii to larger values ( $\sim 12-13$ km) consistent with the prior, but did not significantly improve the Bayesian evidence or resolve the degeneracy issues.

Significance and Claims
The authors position this work as an important next step in the PPM of AMPs, demonstrating the application of the technique to high-quality NICER data. The paper claims that:

Model Complexity is Necessary: A single hotspot model is insufficient for J1808; a two-hotspot geometry is required to explain the pulse profiles.
Disc Modelling Deficiencies: Current simple accretion disc models (blackbody + Gaussian line) are inadequate to capture the spectral features of AMPs, specifically the broadened reflection lines around 1 keV.
Limitations of Flexible Backgrounds: While marginalised backgrounds provide excellent statistical fits, they introduce a degeneracy where the origin of non-pulsed radiation (disc vs. hotspot) cannot be disentangled. Consequently, the inferred mass and radius are not robust without tighter physical constraints on the background.
Future Requirements: The paper concludes that robust parameter constraints for AMPs will require more accurate accretion disc and hotspot modelling. Specifically, the inclusion of higher-energy data and polarimetric data (e.g., from IXPE) is expected to provide complementary constraints on inclination, hotspot colatitude, and size, breaking the current degeneracies.

The authors explicitly state that the reported parameters are not final and are subject to refinement pending improvements in the physical modelling of the accretion disc and hotspot geometry.

1. The "One Spot" Theory Didn't Work

2. Adding a Second Spot Helped, But Not Enough

3. The "Flexible Background" Trick

4. The Big Problem: The "Who Did It?" Confusion

5. What They Learned and What's Next

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