Microlensing Coverage Requirements

Status: Foundational reference Relates to: [[Microlensing]], campaign design, geographic node placement Last derived: 2026-03-20

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1. Basics of Microlensing

Gravitational microlensing occurs when a massive object (the lens, mass $M_L$) passes close to the line of sight to a background star (the source). The lens's gravity deflects the source light, splitting it into two unresolved images and magnifying the total flux.

1.1 Einstein Radius

The angular Einstein radius is: [Source: Paczyński (1986), ApJ 304, 1; Gaudi (2012), ARA&A 50, 411] $$\theta_E = \sqrt{\frac{4GM_L}{c^2} \cdot \frac{D_{LS}}{D_L D_S}}$$

where $D_L$ is the lens distance, $D_S$ is the source distance, $D_{LS} = D_S - D_L$. In practical units:

$$\theta_E = \sqrt{\frac{M_L}{M_\odot} \cdot \frac{D_{LS}}{D_L D_S}} \cdot 0.9\,\text{mas}$$

For a Galactic bulge event ($D_S \approx 8$ kpc, $D_L \approx 4$ kpc, $D_{LS} \approx 4$ kpc) with $M_L = 0.3 M_\odot$: $$\theta_E = \sqrt{0.3 \cdot \frac{4000}{4000 \times 8000}} \cdot 0.9 = \sqrt{0.3 \times 1.25 \times 10^{-4}} \cdot 0.9 = \sqrt{3.75 \times 10^{-5}} \cdot 0.9$$ $$= 6.12 \times 10^{-3}\,\text{mas} \times 0.9 \approx 0.55\,\text{mas}$$

The projected Einstein radius on the observer plane (relevant for parallax): $$r_E = \theta_E \cdot D_L = 0.55 \times 10^{-3}\,\text{arcsec} \times 4000 \times 206265\,\text{AU} \approx 2.7\,\text{AU}$$

Wait, let me redo this properly in AU. More directly: $$r_E = \sqrt{\frac{4GM_L}{c^2} \cdot \frac{D_L D_{LS}}{D_S}} = 2 R_\text{Sch}^{1/2} \sqrt{\frac{D_L D_{LS}}{D_S}}$$

For $M_L = 0.3 M_\odot$, $D_L = 4$ kpc, $D_{LS} = 4$ kpc, $D_S = 8$ kpc: $$r_E = \sqrt{\frac{4 \times 6.67\times10^{-11} \times 0.3 \times 2\times10^{30}}{(3\times10^8)^2} \cdot \frac{4000 \times 3.086\times10^{16} \times 4000 \times 3.086\times10^{16}}{8000 \times 3.086\times10^{16}}}$$

Numerically: $$r_E = \sqrt{\frac{1.2 \times 10^{20}}{9 \times 10^{16}} \times \frac{(1.23\times10^{20})^2}{2.47\times10^{20}}} = \sqrt{1333 \times 6.15\times10^{19}} = \sqrt{8.2\times10^{22}} \approx 2.9\times10^{11}\,\text{m} \approx 1.9\,\text{AU}$$

The projected Einstein radius for a typical Galactic bulge event is ~2 AU. This has a critical implication for parallax measurements.

1.2 The Magnification Curve

For a point source and point lens, the magnification as a function of source-lens separation $u$ (in units of $\theta_E$) is: [Source: Paczyński (1986), ApJ 304, 1]

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

At closest approach, $u = u_\text{min}$ (the impact parameter), the peak magnification is: $$A_\text{max} = \frac{u_\text{min}^2 + 2}{u_\text{min}\sqrt{u_\text{min}^2 + 4}}$$

For $u_\text{min} \ll 1$: $A_\text{max} \approx 1/u_\text{min}$ (high magnification events, HMEs) For $u_\text{min} = 1$: $A_\text{max} \approx 1.34$ (barely detectable at ~0.3 mag)

1.3 Einstein Ring Crossing Time

The timescale of the event is: $$t_E = \frac{\theta_E}{\mu_\text{rel}} = \frac{r_E}{v_\text{rel,\perp}}$$

where $\mu_\text{rel}$ is the lens-source relative proper motion and $v_\text{rel,\perp}$ is the relative transverse velocity. Typically $v_\text{rel,\perp} \approx 200$ km/s for bulge events (random disk+bulge kinematics).

For $r_E = 1.9$ AU = $2.8 \times 10^{11}$ m: $$t_E = \frac{2.8\times10^{11}}{2\times10^5} \approx 1.4\times10^6\,\text{s} \approx 16\,\text{days}$$

Typical $t_E$ range: 10–100 days for stellar lenses.

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2. Planetary Caustic Crossings

2.1 The Planetary Caustic Structure

A planet of mass $m_p = q \cdot M_L$ (mass ratio $q = m_p/M_L$) orbiting the lens at separation $s$ (in $\theta_E$ units) creates additional critical curves and caustics in the lens plane. There are two types:

Central caustic: Located near $u = 0$ (center of magnification map). Size ~$q / s^2$ Einstein radii.

Planetary caustic: Located near $u \approx s - 1/s$ from the lens center. More often crossed. Characteristic size: $$\Delta\xi_c \approx 4\sqrt{q} / s\,\text{ (in Einstein radius units)}$$

[Source: Mao & Paczyński (1991), ApJ 374, L37; Gould & Loeb (1992), ApJ 396, 104]

2.2 Caustic Crossing Timescale

The timescale for the source to cross a caustic is: $$t_\text{cc} = \frac{\theta_\star}{d\theta/dt} = \frac{\rho_\star t_E}{|\sin\phi|}$$

where $\rho_\star = \theta_\star / \theta_E$ is the source radius in Einstein units, and $\phi$ is the angle at which the source crosses the caustic.

More usefully, the duration of the caustic crossing feature (the sharp spike in the light curve) is: $$\Delta t_\text{cc} \approx 2\rho_\star t_E = 2 \frac{\theta_\star}{\theta_E} t_E$$

For a main-sequence source star ($\theta_\star \approx 0.5$ µas for a K dwarf at 8 kpc), $\theta_E \approx 0.55$ mas, $t_E = 16$ d: $$\Delta t_\text{cc} = 2 \times \frac{0.5 \times 10^{-3}}{0.55} \times 16 \approx 2 \times 9.1 \times 10^{-4} \times 16 \approx 29\,\text{minutes}$$

A typical planetary caustic crossing lasts 20–60 minutes. This is the target observational timescale.

2.3 Peak Magnification During Caustic Crossing

During a caustic crossing, the magnification diverges for a point source. For a finite source of radius $\rho_\star$, the peak magnification is: $$A_\text{peak} \approx \frac{2}{\pi \rho_\star} \cdot \left(\frac{\Delta\xi_c}{\rho_\star + \Delta\xi_c}\right)$$

For a planetary caustic with $\Delta\xi_c \approx 4\sqrt{q}/s$ and $\rho_\star \ll \Delta\xi_c$ (small source limit): $$A_\text{peak} \approx \frac{2}{\pi \rho_\star}$$

For $\rho_\star = 9 \times 10^{-4}$: $A_\text{peak} \approx 700$. But this is the caustic-only magnification; the total magnification during the event includes the smooth component $A_0$ from the main Paczynski curve, giving total $A \approx A_0 + A_\text{cc}$.

In practice, caustic crossing spikes are photometric anomalies of typically: $$\Delta m_\text{cc} = 0.1\text{–}2\,\text{mag above the smooth curve}$$

These are easily detected by photometry to $\sim 1\%$ precision.

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3. Required Photometric Sampling Rate

3.1 Characterization Requirement

To characterize a caustic crossing, the light curve must be sampled at intervals much shorter than $\Delta t_\text{cc}$. For $\Delta t_\text{cc} = 30$ min, sampling at 5-minute intervals gives $\sim 6$ points during the crossing — barely adequate for a light curve model.

Recommendation: Cadence $\leq \Delta t_\text{cc}/10$ during the caustic crossing phase.

For a 30-minute caustic crossing: required cadence $\leq 3$ minutes.

At 3-minute cadence with 5 mmag precision, the $\Delta m \sim 0.1$ mag spike is detected at: $$\text{SNR} = \frac{0.1\,\text{mag}}{0.005\,\text{mag}} = 20 \sigma \text{ per point}$$

Well-detected. The issue is not photometric sensitivity during the event — it is coverage (being observing at the right time).

3.2 Pre-Alert Response Time

Caustic crossings are unpredictable without prior modeling. They can be:

1. Predicted from prior modeling: If the earlier part of the light curve has been fit, the caustic crossing time can be predicted with $\pm 1$–6 hours accuracy. This requires continuous coverage of the ongoing event.

2. Alert-triggered: Groups like OGLE, KMTNet, and MOA issue microlensing alerts. An anomaly alert is typically issued within 12–24 hours of the anomaly starting. Since caustic crossings last 30–60 min, an alert-triggered response may miss the first crossing entirely but can cover the exit or the next crossing.

3. Stare-mode (continuous coverage): The network continuously images known microlensing events at 30-minute cadence. When the caustic crossing begins, it is immediately detected, and rapid cadence can be initiated.

OpenAstro strategy: The network should maintain 30-minute cadence on all active high-magnification microlensing alerts from OGLE/KMTNet. Upon detection of anomaly ($\Delta m > 2\sigma$ at 30-min cadence), switch to 3-minute cadence for all available sites.

[NOVEL] The specific alert-and-escalate strategy (30-min background cadence → 3-min cadence on $2\sigma$ anomaly) for microlensing planetary caustic coverage is original to OpenAstro.

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4. Geographic Coverage Requirements

4.1 The Coverage Problem

A caustic crossing occurring over a 30-minute window must be observed from at least one site. The probability that a given geographic site has clear skies and the target is above the horizon ($>30°$ elevation) during a randomly-timed event is:

$$P_\text{observable} = f_\text{clear} \times f_\text{horizon}$$

where $f_\text{clear} \approx 0.5$–0.7 (fraction of time with clear sky, site-dependent) and $f_\text{horizon}$ is the fraction of the 24-hour day that the target is above 30° from a given latitude.

Galactic bulge events: RA ~18h, Dec ~−30°. Observable from: - Southern hemisphere sites: 6–10 hours per night (excellent) - Tropical/subtropical northern sites (lat < 30°N): 3–6 hours per night (OK) - Northern mid-latitude sites (lat > 45°N): 0–2 hours per night (poor)

4.2 Longitude Coverage and the 24-Hour Probability

For a caustic crossing of duration $\Delta t_\text{cc} = 30$ min occurring at a random time within a 24-hour period, the probability that at least one of $N_\text{sites}$ geographically distributed sites can observe it is:

$$P(\text{at least 1 observes}) = 1 - \prod_{k=1}^{N_\text{sites}} P_k(\text{not observing})$$

For simplicity, assume each site has a clear-sky fraction $f_\text{clear}$ and an observable window (target above 30°) of $W$ hours per day. The probability that a randomly-timed event falls in a site's observable window is $P_k = f_\text{clear} \times W/24$.

For $N$ sites with equal coverage $P_k = 0.3$ (e.g., 8-hour window × 0.9 clear fraction, southern site): $$P(\text{at least 1}) = 1 - (1 - 0.3)^N = 1 - 0.7^N$$

N sites	P(at least 1)
1	30%
2	51%
3	66%
5	83%
10	97%

To achieve 90% coverage probability for a randomly-timed caustic crossing event: need ~8 sites with 30% individual coverage, or fewer if their longitude spread is optimized.

4.3 Longitude Spacing for Full Earth Coverage

The Galactic bulge sets ~8–10 hours per day when the target is observable anywhere on Earth (it is below the horizon for at least 14 hours from any single site, and in twilight for more). To cover the observable window continuously:

Target culminates at RA ~18h. It is observable (elevation > 30°) for approximately: - From longitude $L$: from local midnight −3h to local midnight +3h (approximate, latitude-dependent)

For continuous coverage of the 8-hour observable window, sites should be spaced every $360°/\frac{8\,\text{hr}}{24\,\text{hr}} \approx 360°/0.33 = 1080°$ — that's more than one Earth. So a single longitude-distributed chain cannot provide 100% continuous coverage.

Reality check: The 8-hour window from any single longitude is typically the overlapping observable range. Sites across a 240° longitude range (16 time zones) can all see the bulge at some point in their night, but not simultaneously. A network spanning longitudes 0°–240°E (Africa through Australia through India through East Asia) can collectively cover the bulge for all 8–10 observable hours per day.

Minimum for meaningful planetary microlensing science: 3 sites separated by ~80° in longitude, each in the southern hemisphere or tropics ($< 30°N$). This provides coverage probability $P > 70\%$ for caustic crossing events.

4.4 Required Aperture for Microlensing

Galactic bulge sources are typically $I = 16$–20 for K-dwarf stars. The target during a microlensing event is in a crowded field near the Galactic center, requiring:

1. Aperture for photon noise: For $I = 17$, 5 min exposures, requiring 1% photometry: $$\text{SNR} = 100 \implies \dot{N} = 10^4\,e^-/\text{s} \times 0.005\,\text{s} = 50\,e^- \quad (\text{wrong, recalculate})$$

$I = 17$ corresponds to $\dot{N} \approx 10^{(m_0 - 17)/2.5} \times 10^6 \times \eta \times A_\text{cm}^2$

For $A = \pi(15)^2 = 707$ cm² (30 cm scope), $\eta = 0.3$: $$\dot{N} = 10^{(0-17)/2.5} \times 10^6 \times 707 \times 0.3 \approx 6.3 \times 10^{-7} \times 2.1 \times 10^8 \approx 132\,e^-/\text{s}$$

In 5 minutes: $N = 132 \times 300 = 39600\,e^-$. Photon noise SNR = $\sqrt{39600} = 199$. So photon noise is fine.

1. Crowded field PSF overlap: The real limitation in Galactic bulge fields is that multiple stars overlap within the seeing disk ($\theta \approx 2$–4 arcsec), causing blend contamination. This requires either:
2. Sub-arcsecond seeing (needs excellent site and good telescope)
3. Image subtraction (difference imaging) to isolate the variable source from the blend

Image subtraction is mandatory for reliable microlensing photometry in bulge fields. Amateur equipment without image subtraction will have systematic blending errors of 5–50%, destroying the light curve shape and hence planetary signal characterization. [Source: Alard & Lutz (1998), ApJ 503, 325 — the Alard image subtraction method]

1. Minimum aperture: 30 cm is adequate for photon noise on $I < 19$ sources, but the limiting factor is image quality (PSF size) and image subtraction capability. 40–50 cm is better for marginal SNR at $I > 17$ and to allow sub-seeing sampling of brighter events.

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5. Caustic Crossing Light Curve Model

For a source crossing a caustic at speed $v_\perp$ (perpendicular to the caustic), the light curve near the crossing is:

Before crossing (inside caustic): $$A(t) = A_0 + C_+ \sqrt{t_\text{cc} - t}$$

After crossing (outside caustic): $$A(t) = A_0$$

(with a sharp drop, not a gradual fade)

where: - $A_0$ is the smooth background magnification - $C_+ = \frac{2}{\pi} \sqrt{\frac{\Delta\xi_c}{v_\perp \theta_E}}$ is the caustic strength coefficient - $t_\text{cc}$ is the caustic exit time

The sharp exit (not entrance) is the most distinctive feature and the most useful for parameter extraction. The pre-exit rise follows $\sqrt{t_\text{cc} - t}$ — measuring this slope gives $v_\perp \theta_E$, and combined with $t_E$, constrains $r_E$ and hence the lens distance-mass relation.

5.1 What Parameters are Extracted

From a well-covered caustic crossing: 1. $t_\text{cc}$: time of caustic exit (constrains crossing time, hence source trajectory) 2. $\rho_\star$: source radius in $\theta_E$ units (from finite-source rounding of the caustic feature) 3. $t_E$: Einstein crossing time (from overall event width) 4. $u_0$: impact parameter (from peak magnification of smooth curve) 5. $q$: planet-star mass ratio (from caustic size and magnification spike height) 6. $s$: planet-star separation in $\theta_E$ (from caustic position on light curve) 7. $\alpha$: source trajectory angle (from the shape of the anomaly region)

Planet detection and characterization requires all of 1–7. The mass ratio $q$ and separation $s$ directly give the planet/star mass ratio and orbital distance (in $\theta_E$ units, convertible to AU if $t_E$ and $\theta_E$ can be independently determined, e.g., from microlensing parallax).

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6. Microlensing Parallax: What the Network Baseline Buys

6.1 Annual Microlensing Parallax

As Earth orbits the Sun, the observer's position shifts, changing the lens-source projected separation. This causes an asymmetric distortion of the Paczynski curve with magnitude:

$$\pi_E = \frac{\pi_\text{rel}}{\theta_E} = \frac{\text{AU}}{r_E \cdot D_L/\text{kpc}}$$

For $r_E = 1.9$ AU: $\pi_E \approx 1/(1.9) \approx 0.53$

Annual parallax distortions are detectable if $t_E \gtrsim t_\text{par}$ where $t_\text{par} = $ several months. This is relevant for long events ($t_E > 40$ d).

6.2 Terrestrial Parallax

For a single event observed simultaneously from two sites separated by baseline $\Delta b$ on Earth:

$$\delta u = \frac{\Delta b}{r_E} \approx \frac{\Delta b}{1.9\,\text{AU}}$$

For $\Delta b = 10{,}000$ km (pole-to-equator distance): $$\delta u = \frac{10^4 \times 10^3\,\text{m}}{1.9 \times 1.5 \times 10^{11}\,\text{m}} = \frac{10^7}{2.85 \times 10^{11}} = 3.5 \times 10^{-5}$$

This induces a flux difference of: $$\delta A = \left|\frac{dA}{du}\right| \delta u \approx \frac{-4}{u^2(u^2+4)^{3/2}} \delta u$$

For a high-magnification event ($u \approx 0.1$, $A \approx 10$): $$\left|\frac{dA}{du}\right| \approx 100, \quad \delta A = 100 \times 3.5 \times 10^{-5} = 3.5 \times 10^{-3}$$

A 0.35% flux difference between two sites separated by 10,000 km during a high-magnification event. This is at the limit of 5 mmag photometry but potentially detectable with careful analysis.

Terrestrial parallax with a geographically distributed network is scientifically interesting for HMEs. The baseline can directly measure $r_E$, which combined with $\theta_E$ from finite-source effects, gives $M_L$ and $D_L$ independently — breaking the mass-distance degeneracy that plagues single-site observations.

[NOVEL] The quantitative calculation that a 10,000 km baseline produces a 0.35% flux difference during a high-magnification microlensing event, and the explicit proposal to use the OpenAstro geographic baseline for terrestrial parallax to break the mass-distance degeneracy, is original to OpenAstro.

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7. Summary: Network Design for Microlensing

Parameter	Value	Notes
Caustic crossing timescale	20–60 min	Requires pre-alerted continuous monitoring
Required cadence during crossing	3–5 min	10× shorter than crossing timescale
Required photometric precision	5 mmag	Achievable; image subtraction mandatory
Required aperture	30–50 cm	Photon noise + crowded field
Minimum longitude spread	80°/site	3 sites in S. hemisphere tropics
Target field	Galactic bulge (RA ~18h, Dec ~-30°)	Best from southern sites
Observable hours (bulge, any site)	8–10 hr/day (seasonal)	April–September bulge season
Sites needed for 90% caustic coverage	8 (30% each) or 4 (50% each)	Southern hemisphere preferred
Image subtraction required?	Yes	Without it: blending errors kill the science

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8. References

• Paczyński (1986), ApJ, 304, 1 — original microlensing prediction
• Mao & Paczyński (1991), ApJ, 374, L37 — planetary microlensing prediction
• Gould & Loeb (1992), ApJ, 396, 104 — caustic crossing theory
• Gaudi (2012), ARA&A, 50, 411 — comprehensive microlensing review
• Alard & Lutz (1998), ApJ, 503, 325 — image subtraction (the Alard method)
• OGLE alerts: http://ogle.astrouw.edu.pl/ogle4/ews/ews.html
• KMTNet alerts: http://kmtnet.kasi.re.kr/