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General Relativity

Point Gravitational Lens Simulator

Stars, brown dwarfs, and Jupiters within our Galaxy act as gravitational lenses that intensify more distant objects. The reason that a gravitational lens creates a more intense image is that the lens makes the image of the distant object occupy more area on the sky while preserving the surface brightness. The increase in brightness is therefore proportional to the increase in area on the sky.

As discussed on the gravitational lens page, an object directly behind a point gravitational lens creates a ring, called the Einstein ring, around the star creating the lens. This ring will be present as long as the direct line from us through the star at the center of the lens strikes the surface of the more distant object, and as long as the star at the center of the lens has a radius on the sky that is less than the radius of the Einstein ring. If we move the lens to the side, so the the direct line from us through the lens source no longer strikes the more distant object, the ring splits into two oblong images. The images shrink as the lens moves farther and farther to the side. Eventually one image becomes too small to see, and the other image becomes nearly identical to the image we would see if no gravitational lens were present.

The simulator on this page shows the effect of a point gravitational lens on an object at infinity. The simulator shows two figures. The left-hand figure shows the image of a distant blue circle after it has passed through the gravitational lens. When the page loads, the image will be of an Einstein Ring just on the verge of splitting into two images. In the center of the ring is a black dot that gives the position of the star creating the gravitational lens. The right-hand image shows the image of the blue ball in the absence of a lens, the area of the gravitational lens inside of the radius for the Einstein ring, which is presented as a grey circle, and the position of the star creating the gravitational lens, which is a dark grey dot at the center of the grey circle.

The position of the distant blue sphere is fixed in the simulation. The position of the lens can be changed by moving the mouse pointer over the grey disk in the right-hand figure, and dragging the disk to the desired location by holding down the left-hand mouse button as the mouse pointer is moved.

Figure Caption This simulator shows the image created by a point gravitational lens. The right-hand figure shows the source image, which is a blue circle, the lens source, which is a dark-grey dot, and the region around the lens source that is bounded by the Einstein ring radius. The left-hand figure shows the image of the blue circle after it passes through the gravitational lens. The lens source is given by a black dot in this figure. To move the gravitational lens, place the mouse cursor over the grey disk in the right-hand figure, and while pressing the left-hand mouse button, drag the mouse cursor to a new position. As described below, the lens can also be moved with the keyboard arrow keys.

Simulator Notes

Simulator Results

The simulator shows three basic results: a full Einstein ring, an Einstein ring on the verge of splitting into two images, and a pair of images. The simulation starts with the lens positioned to give an Einstein ring on the verge of splitting. This result occurs when the center of the lens source is placed on the boundary of the blue circle. The full Einstein ring is found whenever the center of the lens source is placed within the boundary of the blue circle. Two separate images are found whenever the center of the lens source is outside of the boundary of the blue circle.

The area of the image of the blue circle is largest when it forms an Einstein ring. When the image is split into two parts, the part closest to the lens source is the smaller. The total area of the image is larger than the area of the unmagnified blue circle when the blue circle is inside the grey disk. As the lens moves away from the position of the blue circle, so that the blue circle lies outside of the grey disk, the area of the image nearest the lens source goes to zero; the image farther from the lens source moves to the position of the blue circle, acquiring the shape of a circle, as the lens moves away.

The two images created by the lens when the lens source is off of the blue circle wrap completely around the lens source when the source is moved onto the blue circle. Of the two initial images, the smaller image, which is closer to the lens source, forms a ring inside of the ring created from the larger image. These two rings join at the Einstein ring radius.

Physics of the Simulator

The blue circle has a radius that is 25% of the Einstein ring radius. If we take our lens source to be a 1 solar mass star at 1 parsec, then the Einstein ring radius on the sky is 0.09 seconds of arc. The blue circle is assumed to be a source at an infinite distance.

Usage

The simulator is very simple to use. Place the mouse cursor over the grey disk of the right-hand figure—the cursor will change when this is done. Press the left-hand mouse button, and while pressing this button, drag the cursor. The grey disk and the dot for the lens source will move as the cursor is dragged, and the image in the left-hand figure created by the lens will change.

The position of the grey disk can be manipulated with the arrow keys of the keyboard once the applet has the focus. The up, down, left, and right arrow keys move the grey disk by a single pixel in the expected direction. Holding down any of these keys causes a series of moves. Pressing an arrow key while pressing the control key causes a 5-pixel jump in the expected direction.

I would appreciate hearing from you if you encounter an error while running the simulator or if you have suggestions for improvement. Send your e-mail to the editor of the website.

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