Gravitational Lens Time Delay Calculator

Introduction

Gravitational lensing changes not only where we see a distant source, but also when we see it. If light from a quasar, supernova, or another compact source passes near a massive foreground object, the mass bends spacetime and creates more than one possible path for the light to follow. Those paths usually have different lengths, and they also pass through different gravitational potentials. As a result, the separate images do not arrive at exactly the same time. The difference between their arrival times is called the gravitational lens time delay.

This calculator estimates that delay for the simplest useful lens model: a point-mass lens. In that model, the lens is treated as if all of its mass were concentrated at one point. Real galaxies are more complicated than that, but the point-mass case is still valuable because it captures the core idea behind lensing delays in a clean analytic form. It is often the best place to start if you want intuition for how mass, alignment, and redshift affect the timing difference between images.

The result is especially relevant in cosmology. Astronomers monitor variable lensed sources and compare the brightness fluctuations seen in each image. If one image brightens first and another follows later, the lag can be measured. With a realistic lens model, that measured delay helps constrain the expansion rate of the universe and the mass distribution in the lens. This page does not attempt a full cosmological reconstruction, but it does provide a fast and physically meaningful estimate of the delay scale for a point-mass lens.

How to Use

To use the calculator, enter three quantities in the form below. The first is the lens mass in solar masses. A value such as 1e11 means one hundred billion times the mass of the Sun, which is a rough galaxy-scale mass. The second input is the dimensionless impact parameter $y$, which describes how far the source lies from perfect alignment when measured in units of the Einstein angle. Smaller values of $y$ correspond to closer alignment. The third input is the lens redshift $z_l$, which accounts here for the usual cosmological time-dilation factor $1+z_l$.

After entering the values, press the compute button. The calculator returns the relative delay $\mathrm{\Delta t}$ in seconds and in days. The seconds value is useful for direct numerical work, while the days value is often easier to interpret because observed strong-lensing delays are commonly discussed on day-to-month timescales. If the inputs are invalid, the calculator shows a short error message instead of a numerical result.

For practical interpretation, keep the following in mind. Increasing the lens mass increases the delay almost linearly because the overall scale contains the factor $\frac{4GM}{c^3}$. Increasing $y$ generally increases the asymmetry between the two image paths and therefore increases the delay. Increasing the lens redshift also increases the observed delay through the multiplicative factor $1+z_l$. The sample table below the explanation gives a quick sense of scale for three representative values of $y$ at fixed mass and redshift.

Formula

In the thin-lens approximation, the arrival time of an image at angular position $\theta$ relative to the unlensed source position $\beta$ is described by the Fermat potential. The dimensionless arrival-time surface can be written as

Formula: τ = 1 / 2 (θ-β)^2 − ψ (θ)

$\tau =\frac{1}{2}(\theta -\beta )^2-\psi (\theta )$

For a point-mass lens, the scaled lensing potential is logarithmic, so $\psi (\theta )=ln\left|\theta \right|$. Solving the lens equation for the two image positions and subtracting their arrival times gives a closed-form expression for the relative delay. Using the dimensionless source position $y$, the observable delay is

Formula: Δt = (4 G M) / c^3(1 + z l)[y sqrt(y^2 + 4) + ln (sqrt(y^2 + 4) + y) / (sqrt(y^2 + 4) − y)]

$\mathrm{\Delta t}=\frac{4GM}{c^3}(1+z{l}_{})[y\sqrt{y^2+4}+ln\frac{\sqrt{y^2+4}+y}{\sqrt{y^2+4}-y}]$

This expression combines two physical effects. One part is the geometric delay, which comes from the fact that the bent light rays travel along paths of different effective length. The other part is the gravitational or Shapiro delay, which comes from the slowing of light propagation in a gravitational potential. The logarithmic term is a signature of the point-mass potential, while the prefactor sets the overall timescale. In the calculator, the constants $G$ and $c$ are fixed to standard SI values, and the mass input is converted from solar masses to kilograms internally.

It is also helpful to understand the units. The impact parameter $y$ is dimensionless, and the redshift is dimensionless as well. The mass carries the physical scale. Because the prefactor contains $\frac{\mathrm{GM}}{c^3}$, the final answer naturally comes out in seconds. Dividing by 86,400 converts the result to days, which is the second value shown by the calculator.

Worked Example

Suppose you want a quick estimate for a galaxy-scale lens with mass 1e11 solar masses, source impact parameter 0.5, and lens redshift 0.5. These are the default values already loaded into the form. When you press the compute button, the calculator evaluates the point-mass delay formula and returns a result on the order of several tens of days.

That outcome makes physical sense. A mass of ${10}^{11}$ solar masses is large enough to produce substantial lensing, and $y=0.5$ means the source is fairly close to alignment, so two images form with noticeably different travel times. The redshift factor $1+z_l=1.5$ stretches the observed delay further. In observational terms, a delay of this size is entirely plausible for a lensed quasar monitored over weeks or months.

You can also use the example to build intuition. If you keep the same mass and redshift but reduce $y$ toward zero, the source approaches near-perfect alignment. The image configuration becomes more symmetric, and the relative delay shrinks. If instead you increase $y$ to 1 or beyond, the asymmetry grows and the delay becomes larger. Likewise, if you multiply the mass by ten, the delay scale increases by roughly a factor of ten. These trends are exactly what the formula predicts.

Representative Values

The following table is filled automatically by the page script for a fixed lens mass of 10¹¹ solar masses and lens redshift 0.5. It shows how the delay changes as the dimensionless impact parameter varies. The values are intended as a quick reference, not as a substitute for entering your own parameters in the calculator form.

Interpretation and Assumptions

This calculator is best understood as an educational and order-of-magnitude tool. It assumes a single isolated point-mass lens, the thin-lens approximation, and the standard analytic expression for the delay between the two images produced by that lens. Those assumptions are often good enough to understand scaling behavior, but they are not a full model of a real strong-lensing system.

In actual observations, the lens is usually a galaxy or galaxy cluster with an extended mass profile rather than a true point mass. Real systems may include dark matter halos, stars, gas, ellipticity, external shear from neighboring structures, and line-of-sight mass contributions. Those effects can shift image positions, change magnifications, and alter the delay relative to the simple point-mass prediction. For precision cosmology, astronomers therefore fit more detailed lens models to imaging, spectroscopy, and time-series data.

Another important assumption is that the calculator uses the compact point-mass formula directly, with the lens redshift entering through the factor $1+z_l$. In a full cosmological treatment, time delays also depend on angular-diameter distances between observer, lens, and source. Those distances are what make measured delays so useful for constraining the Hubble constant. Because this page is designed for simplicity and speed, it does not ask for a source redshift or a cosmological model, and it does not compute a full time-delay distance.

Limitations

The main limitation is that a point-mass lens is an idealization. It is excellent for learning and for rough scaling estimates, but it should not be used as a final scientific model for a real lensed quasar or supernova without further analysis. If you need publication-grade results, you should use a lens model that includes the actual mass profile, image geometry, and cosmological distances inferred from the observed system.

The calculator also expects physically sensible positive inputs. The mass must be greater than zero, the impact parameter $y$ must be positive, and the lens redshift cannot be negative. Very small values of $y$ correspond to near-perfect alignment, where the two images become nearly symmetric and the delay can become small. Very large masses or extreme parameter choices can produce large outputs, but those values still reflect the assumptions of the simple model rather than the full complexity of nature.

Even with those caveats, the calculator remains useful. It shows clearly how gravity can bend light and stretch time in a measurable way. By experimenting with the inputs, you can see how lens mass sets the overall scale, how alignment controls asymmetry, and how redshift affects the observed delay. That intuition carries over to more advanced lensing studies, where the same physical ideas appear inside richer and more realistic models.