Going Further 12.2: The Lens Equation

Here we will go into greater detail on the geometry of a lensed system and derive several results that we use in this chapter. As described in Section 10.2.1, if a gravitational lens did not deflect the light from a source, then the observer would see the source at its true position. However, due to the deflection of the light caused by the lens’s gravity, the observer sees an image (or images) of the source instead.

Figure B.12.3 presents a more detailed schematic of the distances and angles involved in a lensed system. As before, the distance from the observer to the source is D_SO, the distance from the observer to the lens is D_LO, and the distance from the lens to the source is D_LS. The angle of deflection is α and the impact parameter is b. The (imaginary) line between the lens and the observer is the optical axis. The angular separation of the image from the optical axis is θ. In addition, we have drawn in two more angles, which we will call x and y. The angle that light from the source would make with the optical axis if the lens was not there is x. We will use y to label the difference between θ and x.

Figure B.12.3. The figure shows the basic elements of a gravitational lens system. The observer is at the point O; the source is at S. These two are separated by a distance D_SO. The deflection of the light from S makes the image appear to be at position I as seen by the observer. The source has thus been offset by a small amount compared to the case if no lens were present. The lens is taken to be somewhere along the path of the light a distance D_LO from the observer and D_LS from the source. These two distances do not have to be the same. D_SI is the distance between the source and image (in the plane of the sky).Credit: NASA/SSU/Aurore Simonnet

From the diagram we see that the relationship between x, y, and θ is:

\begin{equation} x +y = \theta \end{equation}

Or, written another way:

\begin{equation} x = \theta - y(\theta) \end{equation}

This is called the lens equation. It describes the essential geometrical aspects of a gravitational lens. The angle x is the angle between the lens and the source. The angle y is the angle between the source and the image, and the angle θ is the angle between the lens and the image. The angle y will change as θ changes, and we have made that explicit in the equation with the use of the parentheses to mean y is a function of θ.

We can use the small angle formula to relate the various angles in the diagram to each other and to the distances between the lens, source, and observer. From the diagram we can see that the distance between the source and the image (in the plane of the sky) can be expressed in terms of both the deflection angle, α, and the angle y between the source and image:

\begin{equation} D_{SI}=\alpha D_{LS}=yD_{SO} \end{equation}

In making the second assertion (for the angle y), we have used the fact that the distance from the lens to the source (in the plane of the sky) is much smaller than the distance from the observer to the lens or the source. The diagram is not to scale and so is misleading in this regard.

We should note that in using the small-angle formula we have assumed the space is Euclidean. That is generally not the case. In curved spacetime it is not clear what any of these distances actually is. However, we are free to define the distances such that they satisfy the lens equation and small-angle formula. We can then use our relationship to replace the angle y in the lens equation:

\begin{equation} x = \theta - \alpha \frac{D_{LS}}{D_{SO}} \end{equation}

The angle θ is the separation angle between the lens and the images and is directly observable. Recall that the deflection angle α is related to distribution of mass in the lens and on how far from the lens the light would pass were it not deflected, the impact parameter (b). Therefore the lens equation relates an observable angle to the mass of the lens, whether that mass is observable (via emission of light) or not.

In all cases of astrophysical interest, the spatial extent of the lens will be much smaller than the distance between the lens, source, and observer. In that case the lens can be thought of as a thin sheet of mass, and the total mass of the lens can be thought to be projected onto a plane centered on the lens and perpendicular to the line of sight (this is in keeping with our previous geometrical model assumptions). Assuming for simplicity that the lens is spherically symmetric, the deflection angle α is given by:

\begin{equation} \alpha(b)=\frac{4GM(b)}{c^2b} \end{equation}

where G is the gravitational constant, c is the speed of light, and b is the impact parameter. M(b) is the amount of projected mass within a circle of radius b centered on the lens. It is through this relation that gravitational lensing provides a means of determining the mass of the lensing object. When the lens is a galaxy or cluster of galaxies, this method gives a completely independent mass determination from the dynamical methods discussed in Chapter 7. For galaxies or clusters of galaxies, the dependence on mass can be more complicated than for a point mass, but the basic method is the same.

We can rewrite the lens equation explicitly in terms of the mass of the lens if we substitute our expression for α(b) into it. We get:

\begin{equation} x=\theta-\left(\frac{4GM(b)}{c^2b}\right)\left(\frac{D_{LS}}{D_{SO}}\right) \end{equation}

Or, since from the small angle formula b = θD_LO,

\begin{equation} x=\theta-\left(\frac{4GM(b)}{c^2\theta}\right) \left(\frac{D_{LS}}{D_{LO} D_{SO}}\right) \end{equation}

In this form, the dependence on mass of the lens geometry is more explicit. If we multiply through by θ and move all terms onto one side of the equal sign, we get a quadratic expression:

\begin{equation} \theta^2-x\theta-\left(\frac{4GM(b)}{c^2}\right)\left (\frac{D_{LS}}{D_{LO} D_{SO} }\right)=0 \end{equation}

When x = 0, the source lies on the optical axis of the system, directly behind the lens from the point of view of the observer. In that case the equation simplifies to become:

\begin{equation} \theta^2=\left(\frac{4GM(b)}{c^2}\right)\left(\frac{D_{LS}}{D_{LO}D_{SO}}\right) \end{equation}

This has the solution of the Einstein radius θ_E:

\begin{equation} \theta_E=\sqrt{\left(\frac{4GM(b)}{c^2}\right)\left(\frac{D_{LS}}{D_{LO}D_{SO}}\right)} \end{equation}

In other words, when the source, lens, and observer are all lined up, the observer sees an Einstein ring with an angular size equal to the Einstein radius.

We can simplify the lens equation if we write it in terms of the Einstein radius:

\begin{equation} \theta^2-x\theta-\theta_E^2=0 \end{equation}

In general there are two solutions for this equation, each corresponding to a separate image. These are given by the quadratic formula:

\begin{equation} \theta=\frac{x\pm\sqrt{x^2+4\theta_E^2}}{2} \end{equation}

We can explicitly name these two solutions as follows:

\begin{equation} \theta_+=\frac{x+\sqrt{x^2+4\theta_E^2}}{2} \end{equation}

\begin{equation} \theta_-=\frac{x-\sqrt{x^2+4\theta_E^2}}{2} \end{equation}

We discuss the consequences of these two solutions in Section 12.2.2.