Astronomy · Evidence for planets around other stars

Direct imaging and gravitational microlensing

Distinguish photons supplied by a resolved companion from background light magnified by foreground gravity, and examine the different model limits of these two approaches.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

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After reading, you can

Explain the separation and contrast problem in direct imaging.
Describe gravitational microlensing without requiring detected light from a planet.
Read a blended single-lens reference calculation without mistaking it for planetary evidence.

Direct imaging separates a source from the host’s light

Direct imaging seeks light from a companion at an angular position distinguishable from its host star. The companion may reflect stellar light, emit thermal radiation or supply both in the observing band. Direct describes the origin of detected source light; it does not promise a resolved photograph of a planetary surface.

Angular separation and brightness contrast are different obstacles. Instrumental diffraction and the stellar point-spread pattern spread host light across nearby positions. Suppressing and modelling that light can help identify a faint source, but the remaining image must still be assessed for instrumental artefacts.

A measured sky separation is a projected angle at one epoch. Converting it to a physical projected distance needs distance to the system. It does not directly give a three-dimensional separation or the semimajor axis of a relative orbit.

An imaged flux also does not supply a mass automatically. Temperature, radius, atmospheric properties, age and evolutionary assumptions can affect the relation between emitted light and mass. Reflected-light interpretation has its own illumination, geometry and scattering requirements.

A nearby faint point needs a companionship test

A field source can appear close to a star by chance. Position changes over several epochs can test whether the candidate shares the host’s motion or follows a plausible bound orbit, against a background-source alternative. Calibrated astrometry, parallax and the time baseline matter.

Spectra and measurements in complementary bands can test whether the source properties are consistent with the proposed companion model. A consistent colour is useful evidence, but it is not by itself a unique mass or a proof of gravitational binding.

Image processing can alter signal strength and apparent structure. A detection assessment must account for the method’s response to an actual source, residual stellar structure and the region searched. Two reductions of shared exposures are not two independent observing events.

The physical question is whether the combined astrometric, photometric and other evidence supports a particular companion interpretation. Direct imaging adds access to companion photons while retaining an inference problem about what produced them.

Microlensing records another star’s changing light

Gravitational microlensing occurs when a foreground mass changes the paths and apparent magnification of light from a background source as their alignment changes. The small image separations are generally unresolved in the simple photometric case considered here, so the measured record is a changing combined brightness.

A planetary companion to the foreground lens can perturb the magnification pattern. Its own light need not be detected. The source supplying the magnified photons and the foreground system whose gravity is inferred are therefore different objects.

The shape of a planetary perturbation depends on the lens configuration, source trajectory and other parameters. It need not be a second positive spike. Interpreting an anomaly requires appropriate multiple-lens calculations and comparisons with alternative models.

Microlensing can constrain combinations such as a lens mass ratio and projected separation in a defined model. Lens mass, distances and relative motion can remain partly degenerate. An anomaly height or an event duration alone is not a direct planet-mass measurement.

Calculate a reference curve before asking about departures

Use an original ideal point-lens and point-source model with straight relative angular motion. Let u be their angular separation divided by the positive Einstein-angle scale. The magnification is A(u) = (u² + 2)/[u√(u² + 4)]. This formula describes one point lens, with no planet included.

Choose closest separation u₀ = 0.5 at t₀ = 0 and crossing-time unit t_E. Then u(t) = √[0.5² + (t/t_E)²]. Here t_E is the time for the relative angular motion to cover one Einstein angle. It is not an orbital period.

Let the unlensed source supply 100 arbitrary same-band flux units and constant blended light supply another 50. Observed flux is F = 100A + 50, with unlensed baseline 150. Normalized observed brightening is F/150, rather than A itself.

At closest approach A ≈ 2.182821, giving F ≈ 268.2821 units and F/150 ≈ 1.788547. Blended light reduces the normalized brightening relative to source magnification. Both curves are symmetric about zero in this specified model and approach one at large time separation.

Original single-point-lens reference with u₀ = 0.5, source flux 100 and constant blend 50. No planet is present in this calculation.
t / t_E	Source magnification A	Observed F / 150
−2	1.055501	1.037001
−1	1.268670	1.179113
−0.5	1.666667	1.444444
0	2.182821	1.788547
+0.5	1.666667	1.444444
+1	1.268670	1.179113
+2	1.055501	1.037001

A reference brightening with no planet

Scroll sideways to see the full diagram. Its explanation is below.

Original single-point-lens and point-source reference with NO PLANET present. Relative angular motion is rectilinear; closest separation u₀ = 0.5 occurs at t = 0. Time is divided by the Einstein-angle crossing time t_E, rather than an orbital period. Separation is u = √[0.5² + (t/t_E)²]; source magnification is A = (u² + 2)/[u √(u² + 4)]. The unlensed source supplies 100 arbitrary common same-band flux units; unchanged blended light supplies 50, giving an unlensed baseline of 150. Combined model flux is F = 100A + 50, and the plotted normalized signal is F/150. Both analytic curves share a zero-based dimensionless amplitude axis from 0 to 2.5 and time t/t_E from −3 to +3. They are symmetric around 0 and approach 1 far from the event, while remaining above 1 at the finite plotted endpoints. Peak source magnification is about 2.182820625327; peak combined flux is about 268.282062533 units, and F/150 is about 1.788547083551. Constant blended light dilutes the normalized brightening. Solid blue A has circular table markers; dashed orange F/150 has diamond table markers. Dense analytic curves and seven marked table times are synthetic model checks rather than measured samples. The reference includes no additional lens, planetary anomaly, arbitrary bump, finite-source correction, occultation, parallax or lens orbital acceleration. Crossing time alone supplies neither a planetary orbital period nor a unique lens or planet mass.

Different methods leave different unanswered questions

The reference omits finite source size, parallax, lens orbital acceleration, additional lenses and occultation. It cannot be turned into a planetary event by drawing an arbitrary extra bump. A departure calls for a separately justified model and quantitative comparison with the measured data.

The crossing time mixes the angular lens scale with relative proper motion; the lens scale itself depends on mass and distances. Additional constraints can help separate those quantities. A chance alignment is generally not an event that repeats each orbital year, though later observations may still constrain the lens system.

Historical microlensing work, including the Polish-led OGLE programme with observations in Chile and international collaborations, illustrates the importance of monitoring many background sources. This methodological context supplies no current survey status, detection rate or catalogue total.

Direct imaging and microlensing thus obtain different evidence: one separates companion light, the other models gravity’s influence on background light. The next lesson asks how false positives and selection effects affect the interpretation of any resulting candidate list.

Worked examples

Normalize the actual measured sum

Why is the model’s peak observed brightening about 1.788547 rather than 2.182821?

The source is magnified to 100 × 2.182821 ≈ 218.2821 flux units.
Add the unchanged blend to obtain about 268.2821 units.
Divide by the full unlensed baseline 100 + 50 = 150.

Only the source component receives the stipulated magnification. The constant blend dilutes the normalized brightening.

Keep a projected image quantity distinct from an orbit

Does one angular separation in a direct image determine a companion’s semimajor axis?

Distance is needed to convert the angle to a projected physical separation.
The line-of-sight separation and orbital phase are still unknown from that image alone.
A semimajor axis is a parameter of a three-dimensional orbital model constrained through further data.

No. A single projected angular separation is a different quantity from a physical orbital semimajor axis.

Try the reasoning

Must a microlensed planet supply detectable light for its gravity to affect the event?
Show the answer and reasoning
No.
The measured photons can come from the background source whose light is magnified.
Does the original point-lens table provide planetary evidence?
Show the answer and reasoning
No.
Its specified model contains one point lens and no planet.
Is t_E the period of a planet’s orbit?
Show the answer and reasoning
No.
It is an angular crossing time that depends on lens geometry and relative motion.
Does every faint point close to a star in one processed image establish a bound planet?
Show the answer and reasoning
No.
Background sources, residual host light and other alternatives need calibrated multi-epoch and complementary tests.

What to carry forward

Direct imaging obtains companion light while requiring contrast, separation and companionship checks.
Microlensing infers foreground gravity through the magnification of background-source light.
Model parameters, blending and degeneracies must remain explicit before a signal is interpreted as a planetary property.

References

Teaching resources and research supporting this lesson’s explanations and examples.