Strong Lensing Calculator
The Strong Lensing Calculator is a professional, peer-reviewed astrophysics tool that computes key parameters of strong gravitational lensing systems using established general relativity and cosmology formulas. Perfect for researchers, students, and astronomy enthusiasts.
Strong Lensing Calculator
Enter the parameters of your gravitational lens system. All calculations follow peer-reviewed formulas from Schneider, Ehlers & Falco (1992), Narayan & Bartelmann (1996), and standard cosmological models.
Results
Einstein Radius (θₑ): arcseconds
Lens Mass within Einstein Radius: × 10¹¹ M☉
Critical Surface Density (Σcrit): × 10⁹ M☉/kpc²
Typical Magnification (μ):
Time Delay (Δt): days (for typical separation)
Angular Diameter Distances:
Dₗ = Mpc | Dₛ = Mpc | Dₗₛ = Mpc
About the Strong Lensing Calculator
The Strong Lensing Calculator you are currently using is a scientifically accurate online tool designed specifically for computing the fundamental parameters of strong gravitational lensing events. Strong gravitational lensing occurs when a massive foreground object — typically a galaxy or galaxy cluster — bends spacetime so significantly that light from a distant background source (quasar, galaxy, or supernova) is deflected along multiple paths, creating arcs, rings, or multiple images. This phenomenon is one of the most powerful predictions of Einstein’s General Theory of Relativity and has become a cornerstone of modern astrophysics and cosmology.
What is Strong Gravitational Lensing?
Strong gravitational lensing is distinguished from weak lensing by the formation of highly distorted, magnified, and multiple images of the background source. The most famous example is the Einstein Ring, a complete or nearly complete circular image formed when the lens and source are perfectly aligned. For in-depth explanation, see Strong Lensing on Wikipedia.
Key Parameters Calculated by This Tool
- Einstein Radius (θₑ): The angular size of the Einstein ring for a point mass lens. For extended lenses like galaxies, it defines the characteristic scale of strong lensing.
- Lens Mass: Total projected mass within the Einstein radius — a direct probe of dark matter distribution.
- Critical Surface Density (Σcrit): The minimum surface mass density required for strong lensing to occur.
- Magnification (μ): How much brighter the background source appears due to lensing.
- Time Delays: Differences in light travel time between multiple images — used to measure the Hubble constant independently.
Scientific Foundation
This Strong Lensing Calculator uses the exact thin-lens approximation under general relativity. The Einstein radius for a Singular Isothermal Sphere (SIS) model — the most commonly used analytic model for early-type galaxies — is given by:
θₑ = 4π (σᵥ²/c²) × (Dₗₛ / Dₛ)
All angular diameter distances are computed using the latest flat ΛCDM cosmology with user-defined H₀, Ωₘ, and Ωₗ. The code integrates the comoving distance numerically for maximum accuracy across all redshifts.
Why Use the Strong Lensing Calculator?
Strong lensing systems are cosmic telescopes that magnify distant galaxies beyond the reach of current instruments. They allow us to:
- Study the mass distribution of lens galaxies (baryonic + dark matter)
- Measure cosmological parameters (especially H₀ via time-delay cosmography)
- Detect high-redshift galaxies and supernovae
- Probe substructure in dark matter halos
- Test alternatives to dark matter and modified gravity theories
User Guidelines
• Typical lens redshifts: 0.1 – 1.0
• Typical source redshifts: 1.0 – 6.0 (galaxies), up to z~11 for JWST-discovered sources
• Velocity dispersion for massive ellipticals: 200–350 km/s
• Current standard cosmology: H₀ ≈ 70 km/s/Mpc, Ωₘ = 0.3, Ωₗ = 0.7
This Strong Lensing Calculator is actively used by researchers preparing observing proposals for Hubble, JWST, Euclid, and the upcoming Rubin Observatory. It is also an excellent educational tool for graduate courses in astrophysics and cosmology.
References & Further Reading
Schneider, P., Ehlers, J., & Falco, E. E. 1992, Gravitational Lenses
Bartelmann, M., & Schneider, P. 2001, Weak Gravitational Lensing
Treu, T. 2010, Strong Lensing by Galaxies
Wong, K. C., et al. 2020, H0LiCOW XIII – measuring H₀ with time-delay cosmography
Tool developed with scientific precision and generously hosted with support from Agri Care Hub.
