Estimate lensing radius using mass and distance geometry. Check arcseconds, radians, and projected ring size. Use flexible units for quick astronomy homework and analysis.
Angular Einstein Radius:
θE = √[(4GM / c²) × (DLS / (DL × DS))]
Physical Einstein Radius:
RE = DL × θE
This calculator assumes a point-mass lens. It also assumes near alignment of observer, lens, and source. Auto mode uses DLS = DS - DL. Manual mode is better when you already know the lens-source separation term.
| Case | Lens Mass | DL | DS | DLS Mode | Approx. θE | Approx. RE |
|---|---|---|---|---|---|---|
| Stellar Microlens | 1 M☉ | 4 kpc | 8 kpc | Auto | 1.009 mas | 4.036 AU |
| Black Hole Lens | 10 M☉ | 2 kpc | 8 kpc | Auto | 5.526 mas | 11.053 AU |
| Galaxy Lens | 1012 M☉ | 1000 Mpc | 2000 Mpc | Auto | 2.018 arcsec | 9.783 kpc |
The Einstein radius is one of the most useful scales in gravitational lensing. It tells you how large the lensing ring can appear when a distant source, a foreground lens, and an observer line up closely. In simple point-mass lens models, it sets the angular scale for image splitting, ring formation, and microlensing strength.
A larger lens mass usually creates a larger Einstein radius. Geometry also matters. The value depends on the distance to the lens, the distance to the source, and the separation term between lens and source. This is why two systems with the same lens mass can produce different lensing signatures.
This Einstein radius calculator helps students, researchers, and astronomy writers estimate both angular and physical lensing size. The angular result is useful for sky measurements. The physical result is useful for understanding the projected ring scale near the lens plane. Together, they give a clearer view of the lensing setup.
The mass field accepts kilograms, grams, solar masses, Jupiter masses, and Earth masses. Distance fields accept meters, kilometers, astronomical units, light years, parsecs, kiloparsecs, and megaparsecs. This makes the tool flexible for classroom problems, microlensing practice, and galaxy lens examples.
Auto mode uses DLS = DS - DL. This is convenient for quick problems. Manual mode is useful when you already know the lens-source separation term from a prepared model or a textbook exercise. That option makes the calculator more practical for advanced lensing workflows.
The result section shows radians, arcseconds, milliarcseconds, and microarcseconds. It also shows the physical Einstein radius in meters, kilometers, astronomical units, parsecs, and kiloparsecs. This range is useful because stellar lenses often look small in angular units, while galaxy lenses can be easier to understand in arcseconds and kiloparsecs.
It measures the characteristic lensing scale caused by a foreground mass. It is usually expressed as an angle on the sky, but it can also be converted into a physical radius at the lens plane.
More mass bends light more strongly. Because of that, the angular lensing scale grows as mass increases. The relationship is not linear. The radius grows with the square root of the mass term.
The lensing geometry controls how strongly the deflection appears to an observer. Distance to the lens, distance to the source, and lens-source separation all affect the final Einstein radius value.
Use manual mode when your source material already provides the lens-source distance term, or when you want to match a prepared model rather than using the simple difference between source and lens distances.
Yes. It is useful for basic microlensing estimates, especially when you want to compare stellar masses, compact objects, or different observer-lens-source geometries in a quick way.
For stars, solar masses and kiloparsecs are practical. For galaxies, solar masses with megaparsecs often make more sense. Choose units that keep your numbers readable and easy to verify.
Many real lensing angles are tiny. Stellar microlensing often falls in milliarcseconds or microarcseconds. That is normal, and it is why the calculator reports several angular scales at once.
No. It is a strong educational and quick-analysis tool. Detailed cosmology work can require exact angular diameter distances, redshift-based models, and more advanced lens mass profiles.
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.