Model microlensing with point mass or galaxy. Choose units, add velocity, and compute scales instantly. Clear outputs help you validate lensing calculations fast always.
For a point-mass lens, the angular Einstein radius is:
thetaE = sqrt( (4 G M / c^2) * (Dls / (Dl Ds)) )
Here, Dl, Ds, and Dls are angular-diameter distances. The physical Einstein radius in the lens plane is RE = Dl * thetaE.
The critical surface density is:
Sigma_crit = (c^2 / (4 pi G)) * (Ds / (Dl Dls))
| Scenario | Lens mass (Msun) | z_l | z_s | Typical thetaE | Interpretation |
|---|---|---|---|---|---|
| Stellar microlens | 1 | 0.5 | 2.0 | ~0.002 arcsec | Small angle; needs high precision imaging. |
| Galaxy lens | 1e11 | 0.3 | 1.5 | ~1 arcsec | Strong lensing; arcs and multiple images. |
| Cluster lens | 1e14 | 0.4 | 2.0 | ~10-30 arcsec | Large separations; giant arcs possible. |
The Einstein radius sets the characteristic scale of gravitational lensing. For a point-mass lens, it defines where images merge into an Einstein ring when source, lens, and observer align. In microlensing, it controls magnification patterns, event durations, and the size of caustic structures for more complex lenses.
The calculator uses angular-diameter distances Dl, Ds, and Dls. The factor Dls/(DlDs) peaks when the lens is roughly midway to the source, so identical masses can produce very different Einstein angles depending on where the lens sits along the line of sight.
For a 1 Msun lens at cosmological distances, thetaE is often milli-arcseconds or smaller, requiring precise astrometry. Galaxy-scale lenses near 1011 Msun commonly yield arcsecond-scale rings and multiple images. Massive clusters near 1014 Msun can reach tens of arcseconds, creating giant arcs.
The physical Einstein radius RE = Dl thetaE represents the projected size in the lens plane. In microlensing, RE is the natural scale for the source trajectory. Reporting RE in AU or pc helps compare lensing scales to stellar or galactic structures.
The critical surface density Sigmacrit summarizes how much projected mass density is required to produce strong lensing for your geometry. When a lens has surface density comparable to Sigmacrit, image splitting and extended arcs become more likely. Lower Sigmacrit generally means stronger lensing for the same mass distribution.
In redshift mode, distances are derived from an LCDM expansion history with H0, Omega_m, and Omega_Lambda. Small changes to H0 rescale distances and can shift thetaE at the percent level for many cases. For typical work, H0 = 70 km/s/Mpc, Omega_m = 0.3, and Omega_Lambda = 0.7 are reasonable baseline choices.
If you provide a transverse velocity, the tool reports the Einstein crossing time tE = RE/v. For stellar microlensing, tE can range from days to months depending on velocity and geometry. This timescale guides cadence planning for photometric monitoring and follow-up.
A quick consistency check is to compare thetaE to your instrument resolution and compare RE to relevant physical scales. If thetaE is far below resolution, look for microlensing signatures rather than resolved rings. Exported CSV and PDF outputs help you document assumptions and reproduce calculations across multiple targets.
Use angular-diameter distances for D_l, D_s, and D_ls. If you only know redshifts, switch to redshift mode so the tool derives consistent angular-diameter distances automatically.
The lens must lie between observer and source to bend light toward you. If z_s ≤ z_l, the computed D_ls becomes zero or negative, and the Einstein radius is not physical.
Sigma_crit is the surface-density threshold for strong lensing in your geometry. If a lens mass distribution approaches or exceeds Sigma_crit, multiple imaging, rings, and arcs become more likely.
The calculator integrates the standard LCDM distance relation numerically. For near-flat cosmologies, it matches common practice well. If curvature is large, results are approximate and mainly for exploration.
Small thetaE occurs for low-mass lenses, nearby sources, or unfavorable geometry where D_ls/(D_l D_s) is small. Stellar lenses often produce milli-arcsecond or microarcsecond scales.
Use the effective transverse speed of the lens relative to the line of sight, including observer, lens, and source motion. For galactic microlensing, 100–300 km/s is a common order-of-magnitude estimate.
The formula here is for a point-mass scale, which still provides a useful characteristic size. For detailed galaxy models, you would need a mass profile, but thetaE remains a helpful first diagnostic.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.