Map of the Pleiades. This is the map I got from the ``Your Sky'' web site. Yours should be similar.
Deflection of light (not to scale). This shows how light is deflected by a black hole. The top view shows the situation with no black hole present. The star is the grey circle on the right; our point of view from the earth is indicated by the eye symbol on the left. Light goes straight from the star to us. In the bottom view I've added a black hole at the center. The actual path of the light is shown by the solid line; notice the bend as it passes the hole. The dotted line shows where the star appears to be as a result of the hole's effect.
The effect of the black hole is to push the apparent positions of background stars away from the position of the hole. This may seem a paradox; after all, light is attracted to the hole, so you'd expect the images of stars to move in closer. But actually, it works the other way. When you put a massive black hole in front of a distant field of stars, it seems to push the stars away from itself. This is because the light from the star gets bent, so it now comes to us at a bigger angle from the black hole than it did before.
To construct a star-map showing the effect of a foreground black hole, you need to decide how massive the hole is and how far it is from the Earth. I am going to use the symbols M for the hole's mass and D for its distance. In the examples below, I am going to use M = 106 Ms, where Ms is the mass of the Sun, and D = 4000 AU, where 1 AU = 1.5 × 108 km is the average distance from the Earth to the Sun. You can work with these values, or substitute your own choices.
The quantities M and D are used to calculate the ``Einstein angle'' ac using the formula
where G is the gravitational constant and c is the speed of light. This calculation becomes easier once you know that
Using this relation to simplify the first equation, we get
For example, using M/Ms = 106 and D/1 AU = 4000, I get ac = 0.18°.
The next thing you need to decide is where on the star-map to put the hole. It's a little complicated to calculate what happens to the image of a star which is closer than ac to the hole, so it's easiest to put the hole somewhere with no stars closer than this angle.
When you measured the positions of the stars on the chart, you should have also have measured the width of the chart in units of the graph paper squares. Let L be one-half the width of the chart; since the chart is 2° wide, L represents an angle of 1°. For example, the chart I printed out is 5 in wide, and the squares on my graph paper are 0.1 in wide, so I get L = 25.
Now let rc = ac L; this is the radius of the Einstein angle in units equal to the width of your graph-paper squares. (For example, using my values for these numbers, I get rc = 4.5. Your result may be different since you probably have a different value for L.) Using the graph paper as a ruler, draw a circle with radius rc somewhere on the chart so that (1) there are no stars within the circle, but (2) there are stars close to the circle. Plot the location of the hole in the center of this circle.
Finally, you are ready to re-plot each star at the position it will have when shifted by the effect of the hole. Remember that I said the hole's effect is to ``push'' things away from its position. So the game is to move each star away from the hole by a calculated amount. The calculation works like this: using the graph-paper as a ruler, measure the distance of a star from the location of the hole and call the result r. (For example, I placed the hole so that the nearest star on the chart is 5 graph-paper squares away, so for this star I get r = 5.) Then calculate the new distance of the star from the hole, which I will call r1, using this formula:
In other words, the apparent distance of the star from the hole increases by rc2/r. (For example, using rc = 4.5 and r = 5, I get r1 = 5 + 4.5×4.5/5 = 9.05.) To find the star's new position, draw a line r1 graph paper squares long from the hole through the star's old position, and plot the star at the end of this line. Then go on to the next star in the chart. When all the stars have been replotted at their new positions, you have a new star-chart showing how the hole changes the pattern of stars.
Last modified: March 25, 1999