Confirming candidate self-lensing binary detections with radial velocity measurements

The gist The discovery of such systems is a potential method of finding quiescent black holes and neutron stars, most of which will be pristine and unchanged from their formation<ref:2610.30203#pg6>

Introduction to Self-Lensing

Compact objects in approximately edge-on binary orbits with stars are able to produce a self-lensing effect, amplifying light from their companion once per orbit<ref:2610.30203#pg8> This phenomenon is a special case of microlensing where the source and the lens are in a binary system together, in an orbit which is close to edge-on, producing a periodic microlensing amplification of the source once per orbit<ref:2610.30203#pg9> Self-lensing is only possible when the lens is a compact object<ref:2610.30203#pg10> A detection of self-lensing (i.e. periodic microlensing) is de facto a detection of a compact object To date, five self-lensing binaries have been confirmed, each containing a white dwarf lens, all of which have relatively close, short-period orbits However, it is expected that stellar-compact object binaries should span the full range of compact object masses as well as the full range of orbital configurations

Methodology for Parameter Retrieval

The investigation involves simulating detectable self-lensing light curves and using those light curves to generate targeted radial velocity observations, fitting RV jointly with photometry The methods involve several key steps:

1. Generating a library of selflensing photometric simulations covering a range of physical parameters, filtered to include only the simulated light curves which contain visible lensing where the peak amplitude is larger than the photometric noise

2. Performing an MCMC fit against the simulated photometry to sample the posterior distribution of self-lensing physical parameters and find distinct physical solutions for each light curve, assuming stellar properties are known

3. Computing an RV observing schedule using a variable number of observations (between three and ten observations) over a fixed five-year campaign, such that each schedule is optimised to distinguish between the photometric solutions

4. Performing a second MCMC fit constrained jointly by the photometry and RV, using the final chain state from the photometric fit as the initial state for the joint fit

Impact of Radial Velocity Measurements

The study evaluates how many spectroscopic RV measurements are required to accurately retrieve self-lensing system parameters as a function of those parameters The results show that adding RV measurements substantially reduces parameter error, with compact-object class generally easier to determine than the exact lens mass Parameter accuracy is driven mainly by number of RV observations, with RV precision playing a weaker role except for the shortest-period systems This suggests that multi-stage RV follow-up, starting with a small number of observations of many candidates, can efficiently classify self-lensing candidates and identify high-priority systems for further investigation

Parameter Recovery Performance

The paper evaluates the median relative error in recovered lens mass and orbital period alongside compact-object class recovery Figure 4 shows how well the object class is recovered as a function of the number of observations when performing a joint photometric/RV fit The recovery rates for compact-object class span roughly 60–100% depending on number of RV observations, with higher mass lenses being recovered more successfully than lower mass Figure 10 shows the median relative error for fitted lens mass scaled to the detectable population from population synthesis Larger lens masses are recovered with smaller errors than smaller masses, and errors are significantly worse for Nobs < 5, especially for lower lens masses

Conclusion and Recommendations

The recommendation for following up on photometrically detected self-lensing candidates is a staged follow-up strategy A second stage of follow-up observations can then concentrate on the true binaries to ascertain the lens mass to a higher precision The results suggest that the highest mass lenses tend to have the lowest relative errors, even with large orbital periods A targeted radial velocity observing campaign, carefully timing observations to distinguish between possible solutions, is effective in recovering object properties with a minimal number of observations The estimates presented here are optimistic upper limits for confirmed detections under ideal conditions, but bode well for our ability to find self-lensing systems in these and other surveys

Appendix A: Photometric Model Details

The photometric model used includes three photometric effects: self-lensing, ellipsoidal variations, and Doppler boosting The combined model flux is defined as F(t) = F0ASL(t) [1 + AEV(t) + ADB(t)] This makes the simplifying assumption that the ellipsoidal variations and Doppler boosting are of small amplitude The calculation of the source brightness involves integrating the corresponding photon flux over the TESS bandpass, weighted by instrumental response and detector quantum efficiency

Appendix A: Radial Velocity Calculation

The simulated line-of-sight radial velocity is then v r (t) = vtan,(t) sin i - vrad,(t) cos i This reduces to the circular-orbit convention v r (t) ∝ sin φ(t) when e = 0, but for eccentric orbits the radial component means the RV at t0 is not generally zero The simulated line-of-sight radial velocity is then v r (t) = vtan,(t) sin φ(t) − vrad,(t) cos i

Appendix A: Targeted Radial Velocity Scheduling

If the photometric fit only found one plausible solution, three observations are scheduled around this event, one at t0, and two at t0 ± δt This triplet of observations provides information about e and ω If there are multiple plausible photometric MCMC solutions, the predicted RV values for each of them over the observing campaign are calculated, and the difference in RV values for each pair of solutions is calculated The final set of observation times is chosen by iteratively selecting the times which maximise the minimum separation between solutions

Appendix A: Joint Photometric-RV MCMC Fit

The quantity sampled by the MCMC is then the sum of these terms plus the parameter-prior term, ln π(θ) + QRV(θ) + QLC(θ) The reported fitted parameters are posterior summaries rather than single maximum-posterior samples

Appendix A: Population Synthesis Emulator

The emulated class recovery probability for population-synthesis system j is the weighted mean of its neighbours’ class recovery outcomes The final class recovery fraction is the arithmetic mean of these emulated probabilities over the sampled population The emulator estimates the recovery probability for each population-synthesis system from the fitted outcomes of nearby systems in our simulation library

Appendix A: Lens Mass Relative Error

The reported median relative error is the median of the rˆj values for sampled entries within each true lens-mass bin Larger lens masses are recovered with smaller errors than smaller masses, and errors are significantly worse for Nobs < 5, especially for lower lens masses

Conclusion

Combining parameter recovery rates with previous population synthesis results suggests that it is possible to confirm the existence of a substantial number of self-lensing systems in surveys such as TESS, ZTF, LSST, and PLATO<ref:2610

1. Predict compact object class with high confidence using joint photometry and radial velocity measurements. The improved system can distinguish between black holes and white dwarfs/neutron stars by analyzing how RV accuracy affects class recovery, as higher mass objects can be more accurately determined to be such, even if their exact mass cannot be constrained.

2. Develop an optimal, adaptive radial velocity observing schedule for self-lensing candidates. The AI system can compute a schedule that maximises the separation between plausible solutions by using the final set of observation times chosen by iteratively selecting the times which maximise the minimum separation between solutions.

3. Implement a staged follow-up prioritization strategy based on predicted recovery rates. The system can recommend observing targets with fewer observations initially to determine their compact object class, reserving higher precision RV measurements for those that appear to be lower mass objects, as suggested by a staged follow-up strategy would be appropriate, where lower accuracy observations are first obtained for all targets.

4. Accurately estimate lens mass and orbital period using a minimal number of RV measurements. The AI can identify that with only a small number of RV measurements it is possible to determine the broad compact object class in many cases and to measure the lens mass to an acceptable relative error.

5. Automate parameter estimation for long-period systems. The system can handle systems where orbital period can not be determined photometrically by using a targeted radial velocity observing campaign, carefully timing observations to distinguish between possible solutions, is effective in recovering object properties with a minimal number of observations.

Compact objects in approximately edge-on binary orbits with stars are able to produce a self-lensing effect, amplifying light from their companion once per orbit. To date, five white dwarf self-lensing systems are known, while none are known containing black holes or neutron stars. Discovery of such systems is a potential method of finding quiescent black holes and neutron stars, most of which will be pristine and unchanged from their formation. Photometric detections can suffer from degeneracies in the lens mass and orbital parameters when the amplification and crossing time are not well constrained, while photometric searches produce too many false positives to allow radial velocity investigation of every candidate. Therefore we investigate the minimal radial velocity (RV) observational requirements to characterise such systems by simulating detectable self-lensing light curves, using those light curves to generate targeted RV observations, and fitting RV jointly with photometry. We evaluate the median relative error in the recovered lens mass and orbital period, alongside compact-object class recovery. We find that adding RV measurements substantially reduces parameter error, with compact-object class generally easier to determine than the exact lens mass. Parameter accuracy is driven mainly by number of RV observations, with RV precision playing a weaker role except for the shortest-period systems, with even a small number of RV measurements resulting in substantial improvement in accuracy. These results suggest that multi-stage RV follow-up, starting with a small number of observations of many candidates, can efficiently classify self-lensing candidates and identify high-priority systems for further investigation.