Self-Lensing in Binary and Galactic Systems

• Self-lensing is a gravitational phenomenon where the lens and source are members of the same system, as seen in stellar binaries and galactic microlensing.
• In binary systems, periodic brightening pulses enable direct measurement of compact-object mass, radius, and orbital geometry through light curve analysis.
• Self-lensing spans contexts from stellar binaries to massive black hole systems, offering insights into population dynamics, strong-field gravity, and evolutionary scenarios.

Self-lensing denotes gravitational lensing in which the lens and the lensed source are not unrelated line-of-sight objects but members of the same physical system or stellar structure. In stellar-binary usage, a compact object periodically magnifies the light of its luminous companion when the orbit is viewed nearly edge-on; in Magellanic-Cloud microlensing, the term denotes events in which both lens and source belong to the same galaxy; and in the strong-gravity regime of massive black hole binaries it denotes periodic flares produced when one black hole lenses emission from the accretion flow of its companion (Kruse et al., 2014, Mroz et al., 2017, Davelaar et al., 2021).

1. Definitions and domains of use

In the cited literature, self-lensing is used in several related senses, unified by the fact that the lens and source are not independent foreground and background populations. In the stellar-binary case, a compact remnant orbits a luminous star and periodically passes in front of it, producing a repeating microlensing pulse. In the Small and Large Magellanic Clouds, self-lensing means that both lens and source belong to the same external galaxy. In lensing tomography, a related usage appears in the context of source-lens clustering, where the observed source distribution is statistically coupled to the lensing field (Kruse et al., 2014, Mroz et al., 2017, Novati et al., 2011, Yu et al., 2014).

Context	Lens–source relation	Observable
Stellar binaries	Compact object and luminous star in the same binary	Periodic brightening pulse, sometimes with occultation
SMC/LMC microlensing	Lens and source both belong to the same galaxy	Optical depth and event-rate contribution from luminous populations
Massive black hole binaries	One black hole lenses the minidisk of its companion	Periodic self-lensing flares
Lensing tomography	Source sample correlated with the lensing field	Bias in measured shear power spectra

For stellar binaries, the defining scale is the Einstein radius. In ordinary microlensing it is

$$R_E=\sqrt{\frac{4GM_2}{c^2}\frac{D_L(D_S-D_L)}{D_S}},$$

where $D_L$ and $D_S$ are the lens and source distances. In a self-lensing binary, $D_L \approx D_S$ and the distance factor reduces to the orbital separation $a$, so

$R_E \approx \sqrt{\frac{4GM_2 a}{c^2}}.$

This reduction removes the distance degeneracy characteristic of single-pass microlensing and makes the light curve directly sensitive to compact-object mass and orbital geometry (Kruse et al., 2014).

In Magellanic-Cloud applications, the same term does not refer to a bound binary but to microlensing by luminous populations internal to the target galaxy. In the SMC, the lens and source are both members of the SMC itself; in the LMC analyses, the broad “self-lensing” category includes LMC disc and bar stars, the MW disc, and the stellar haloes of both the LMC and the MW, in contrast to MACHOs in the dark haloes (Mroz et al., 2017, Novati et al., 2011).

2. Binary self-lensing physics

A self-lensing binary contains a source star of radius $R_1$, a compact lens of mass $M_2$ and radius $R_2$, and an orbit viewed nearly edge-on. When the compact object crosses the projected stellar disc, two competing effects occur: physical occultation by the lens and gravitational magnification of the source. The net signal is a brightening when lensing dominates occultation, and a dimming when occultation dominates (Kruse et al., 2014, Han, 2016).

For a point lens and point source, the total magnification is

$A(u)=\frac{u^2+2}{u\sqrt{u^2+4}},$

with $D_L$0 the source–lens separation in Einstein-radius units. In actual self-lensing binaries, finite-source effects are usually dominant because $D_L$1, so the pulse is shallow and depends on the ratio $D_L$2. For close binaries observed by TESS, the relevant dimensionless quantities are

$D_L$3

and the peak enhancement in the large-source regime is approximated by

$D_L$4

so larger compact-object mass and larger semi-major axis increase the lensing signal, while larger stellar radius suppresses it (Sajadian et al., 2024).

For KOI-3278-like systems, where $D_L$5, the pulse can be modeled as an “inverted transit” across a limb-darkened stellar disc. The flux during the pulse is

$D_L$6

The term $D_L$7 is the lensing contribution and $D_L$8 is the occultation contribution. When $D_L$9, the net signal is positive (Kruse et al., 2014).

A central complication is the degeneracy between lensing and occultation. For a finite lens deep inside the source disc, the net magnification is approximated by

$D_S$0

and for a fixed observed peak $D_S$1 the corresponding degenerate family satisfies

$D_S$2

This means that different combinations of Einstein radius and physical lens radius can yield nearly indistinguishable light curves, especially for WD lenses with $D_S$3 (Han, 2016).

Self-lensing in X-ray binaries follows the same formal structure. There the Einstein radius is often written

$D_S$4

and the signal is strongest when the angular size subtended by the companion is small, favoring relatively compact companion stars and, in some cases, evolved massive stars such as WR stars. A notable qualitative feature is that the self-lensing signal is stronger in binaries with large separations, which is the opposite of the case for X-ray and radial-velocity techniques (Sorabella et al., 2020).

3. Stellar systems and direct detections

KOI-3278 is the first clear stellar self-lensing binary and remains the canonical example. Kepler photometry revealed a $D_S$5-hour pulse of $D_S$6 amplitude every $D_S$7, accompanied by an occultation half an orbit away. Joint modeling gave a G-dwarf primary with

$D_S$8

and a CO white dwarf with

$D_S$9

in an orbit with

$D_L \approx D_S$0

and Einstein radius

$D_L \approx D_S$1

Because $D_L \approx D_S$2, the event appears as a brightening pulse rather than an eclipse (Kruse et al., 2014).

KIC 12254688 extends the observational record into the TESS era. Two self-lensing pulses were detected in TESS light curves, the first such pulses in that mission, from a long-period F2V+WD binary with $D_L \approx D_S$3 days. A joint self-lensing+RV model yielded, from the Kepler-based fit,

$D_L \approx D_S$4

with

$D_L \approx D_S$5

The pulse amplitude is about $D_L \approx D_S$6 in relative flux, illustrating why TESS can confirm known systems more easily than it can discover them de novo (Sorabella et al., 2024).

PSR J1910−5959A represents a different regime: a NS–WD binary proposed as an observable non-eclipsing self-lensing binary. With

$D_L \approx D_S$7

the predicted peak amplification is $D_L \approx D_S$8–$D_L \approx D_S$9 and the event duration is about $a$0–$a$1 s. The system is notable because it could combine periodic optical amplification with the already observed Shapiro time delay of radio signals, and because the amplification limit can constrain the WD radius (Chan et al., 2024).

4. Surveys, population predictions, and search strategies

Survey work has made self-lensing a population problem rather than a single-system phenomenon. A pilot search in a subset of high-cadence ZTF data identified 12 plausible candidates, but because each candidate was observed to brighten only once, stellar flares or other origins are more likely; the study concluded that periodic recurrence is essential for robust identification (Crossland et al., 2023).

Population synthesis predicts much larger latent samples. One analysis of current and planned large-area optical surveys concluded that they can detect a significant number of self-lensing binaries, $a$2–$a$3s, and emphasized that many observable systems should show multiple flares, which both improves detection and distinguishes them from chance-alignment microlensing (Wiktorowicz et al., 2021). A later study aimed specifically at supernova physics found that, after simple detection criteria including photometric precision and signal-to-noise requirements, predicted rates decrease by approximately two orders of magnitude, but still yield up to a few tens of expected detections for LSST and ZTF in the Galactic disk population; it also found a strong preference for low center-of-mass velocities $a$4 and a model-dependent abundance of mass-gap lenses in the $a$5–$a$6 range (Wiktorowicz et al., 15 Sep 2025).

TESS-specific simulations of detached compact-object binaries show both the promise and the selectivity of survey detection. Using the criteria $a$7 and $a$8 for low-confidence detections and $a$9 and $R_E \approx \sqrt{\frac{4GM_2 a}{c^2}}.$0 for high-confidence detections, the high-confidence efficiencies for detecting WDMS, NSMS, and BHMS systems with $R_E \approx \sqrt{\frac{4GM_2 a}{c^2}}.$1 are $R_E \approx \sqrt{\frac{4GM_2 a}{c^2}}.$2–$R_E \approx \sqrt{\frac{4GM_2 a}{c^2}}.$3, $R_E \approx \sqrt{\frac{4GM_2 a}{c^2}}.$4–$R_E \approx \sqrt{\frac{4GM_2 a}{c^2}}.$5, and $R_E \approx \sqrt{\frac{4GM_2 a}{c^2}}.$6–$R_E \approx \sqrt{\frac{4GM_2 a}{c^2}}.$7, respectively. Detecting lensing-induced features is possible in only $R_E \approx \sqrt{\frac{4GM_2 a}{c^2}}.$8 and $R_E \approx \sqrt{\frac{4GM_2 a}{c^2}}.$9 of detectable WDMS and NSMS events. The predicted numbers of compact companions recovered from the TESS Candidate Target List are $R_1$0–$R_1$1 WDs, $R_1$2–$R_1$3 NSs, and $R_1$4 stellar-mass BH (Sajadian et al., 2024).

Dense stellar environments add a further observational domain. In synthetic globular-cluster populations, present-day clusters contain $R_1$5–$R_1$6 self-lensing sources with $R_1$7, strongly dependent on initial binary fraction, and dominated by WD lenses paired with low-mass main-sequence companions. The predicted populations show bimodal magnitude distributions with peaks at $R_1$8 and $R_1$9 mag at $M_2$0 kpc, and typical Einstein-ring crossing times of $M_2$1 hours. ELT/MICADO surveys of $M_2$2 nearby globular clusters should achieve detection efficiency of $M_2$3–$M_2$4 sources after a year of observations depending on distance and strategy, while multi-year daily-cadence campaigns provide order-of-magnitude improvements (Wiktorowicz et al., 3 Oct 2025).

5. Extragalactic and strong-gravity manifestations

In the SMC, self-lensing means that both the lens and the source star belong to the SMC itself. A three-dimensional model based on Cepheids and RR Lyrae stars, with total stellar mass $M_2$5, reproduces the observed microlensing optical depths if all observed SMC events are due to self-lensing. The predicted mean optical depths are

$M_2$6

$M_2$7

$M_2$8

and the corresponding LSST event rate is $M_2$9–$R_2$0 self-lensing events per year. If the planet frequency in the SMC is similar to that in the Milky Way and an optimized cadence is adopted, the expected extragalactic planet yield is $R_2$1–$R_2$2, i.e. a few SMC planets over the full 10-year survey (Mroz et al., 2017).

Toward the LMC, the broad self-lensing signal from luminous populations can explain the two OGLE-III microlensing candidates without requiring MACHOs. In that analysis, the expected MW disc signal is almost as large as that from LMC stars. The 95% CL upper limit on the halo MACHO fraction is $R_2$3–$R_2$4 for $R_2$5–$R_2$6 in the Bright sample, and below $R_2$7 over this full range in the All sample, showing that luminous self-lensing and MW disc lensing account for the observed rate (Novati et al., 2011).

At the opposite mass scale, self-lensing flares from massive black hole binaries are a strong-gravity realization of the same idea. General-relativistic ray tracing in a superposed binary BH metric shows that a nearly edge-on supermassive binary can produce periodic short-duration flares when one BH lenses the minidisk of the other. For edge-on configurations, the flare apex acquires a distinct central dip associated with the background BH shadow. The phase spacing of the sub-peaks is

$R_2$8

and for the fiducial equal-mass model the inclination window for the dip is

$R_2$9

The follow-up study estimates that about $A(u)=\frac{u^2+2}{u\sqrt{u^2+4}},$0 of the current binary candidates could show this feature, implying a time-domain route to black-hole-shadow measurements in systems that are spatially unresolvable by VLBI (Davelaar et al., 2021, Davelaar et al., 2021).

6. Degeneracies, systematics, and scientific role

The interpretation of self-lensing light curves is limited by parameter degeneracies and by survey systematics. In stellar binaries with finite-size lenses, the lensing–occultation degeneracy is intrinsic whenever both magnification by lensing and de-magnification by occultation occur simultaneously. Example light curves can differ by less than $A(u)=\frac{u^2+2}{u\sqrt{u^2+4}},$1 in fractional flux for distinct $A(u)=\frac{u^2+2}{u\sqrt{u^2+4}},$2 combinations, below Kepler’s practical sensitivity in many cases, which compromises unique inference of compact-object radius and mass from photometry alone (Han, 2016).

A different systematic appears in weak-lensing tomography under the related heading of source-lens clustering. There the observed source density is

$A(u)=\frac{u^2+2}{u\sqrt{u^2+4}},$3

and the coupling between source selection and the lensing field biases the measured E-mode power spectrum. For typical photo-$A(u)=\frac{u^2+2}{u\sqrt{u^2+4}},$4 errors $A(u)=\frac{u^2+2}{u\sqrt{u^2+4}},$5 and redshift-bin size $A(u)=\frac{u^2+2}{u\sqrt{u^2+4}},$6, the effect alters the lensing E-mode power spectrum by $A(u)=\frac{u^2+2}{u\sqrt{u^2+4}},$7–$A(u)=\frac{u^2+2}{u\sqrt{u^2+4}},$8, with $A(u)=\frac{u^2+2}{u\sqrt{u^2+4}},$9 and $D_L$00 of particular relevance. The standard estimator is biased by both intrinsic source clustering and cosmic magnification, whereas the pixel-based estimator suppresses the magnification-induced part but not the intrinsic-clustering part (Yu et al., 2014).

Despite these complications, self-lensing is scientifically distinctive because it is a purely gravitational probe of compact objects in regimes that are otherwise difficult to access. In KOI-3278 it constrains the WD mass, radius, and cooling age and thereby tests CO white-dwarf mass–radius relations and binary-evolution scenarios (Kruse et al., 2014). In compact-object surveys it offers a route to detached, non-accreting NS and BH companions that are inaccessible to X-ray searches, and in population studies it probes long-lived, previously hidden stages of binary evolution that affect double-compact-object merger predictions (Wiktorowicz et al., 2021). In supernova-population studies, the abundance of mass-gap self-lensing systems becomes a diagnostic of remnant-formation physics, with delayed explosion models producing up to $D_L$01 times more $D_L$02–$D_L$03 lenses than rapid models for certain surveys (Wiktorowicz et al., 15 Sep 2025).

Self-lensing is therefore not a single observational phenomenon but a family of lensing configurations tied together by internal source–lens association. Its most mature form is the repeating pulse of a compact binary; its broader uses range from Magellanic-Cloud microlensing to BH-shadow tomography. Across these domains, the common feature is that the geometry is internally constrained rather than accidental, which turns weak, short-lived magnifications into direct probes of compact-object mass, binary structure, stellar populations, and, in some regimes, strong-field gravity.