5. One for Martin Gardner

18th Century Oxford astronomer James Bradley observed a small displacement in the position of stars over the course of the year.  In 1727, he determined that this was caused by, and in the direction of, the motion of the earth around the sun and a fixed speed of light rays.  He further deduced from this that the speed of light was 183,000 miles per second. Displacement of light rays due to motion of the observer is called stellar aberration.

An often used analogy is a man with an umbrella in a rain shower.  If he stands in one spot, he holds the umbrella directly overhead to keep dry.  However, if he starts walking, he must tip the umbrella ahead to keep the raindrops off.  The faster he moves, the more he must tip the umbrella from vertical.

Non-relativistic pluvial aberration

The geometry is simple for slow speeds, but should his motion be relativistic, we would need to use the aberration formula Einstein derived in his 1905 paper.

We have a different analogy to demonstrate aberration, one that can show both slow speed and relativistic aberration.  We place two boats on a lake, our sailboat ‘A‘ doing 6 knots eastward, and a motorboat ‘B‘ on a roughly southwesterly course at 10 knots, and we’re going to collide at point ‘C‘.  The arc is at 10 knots.

The ‘true’ collision angle (Φ phi) off our course is DCB, but we see the motorboat coming at us along line AB, an apparent angle of DAB.  Setting up a simplified trig formula for these angles, we find the true angle DCB is atan(sin Φ/cos Φ).  The apparent angle (α alpha) DAB has our sailboat’s speed added to the cos leg, atan(sin Φ/(cos Φ + v)).

The late Martin Gardner was a mathematical recreationist who often wrote about interesting mathematical oddities, puzzles, and games in his books and in his Scientific American column.  He especially liked ruler and compass solutions to complicated problems, where you construct with only arcs and circles, perpendicular and parallel lines, and a limited number of angles.

The above analogy is a ruler and compass construction.   Here, then, is a ruler and compass solution to finding relativistic stellar aberration for any speed and true stellar angle.

Speeding things up in the diagram example, A is now our starship doing .6 c, and B is a light ray from a distant star that we see when we get to point C.  Line BC actually represents the angle of the field of light rays that we fly through getting to point C.  The arc is scribed at lightspeed, 1.0 c.

Einstein’s equation covers aberration at all speeds, from snails to starships.  The relativistic correction in his formula increases the forward sweep, sharpens the apparent angle off the bow when you get up to relativistic speeds.  To sharpen the apparent angle DAB, we must either shorten the vertical line BD, or lengthen the horizontal line DA.  We can do that by dividing DA by the inverse of the Lorentz factor (0.8 for .6 c).

The question now becomes, where can we find the inverse Lorentz factor for any given speed?  Not coincidentally, if we construct a vertical from our starship to the lightspeed arc, the intersection E is the inverse factor

for any speed, snail through starship.  Now to scale up the line DA.

We construct a line from D through the point E, giving us triangle DAE.  Dividing AE by the inverse Lorentz factor (essentially itself) yields a length of 1.0.  By similar triangles, if we scale AE up to 1.0, the new base will be line DA scaled up by the same factor.

We extend line DE to intersect (point F)a constructed parallel line 1.0 away from the base, then drop a vertical to the base line, intersecting at point GDG is now our corrected base length, and angle DGA is the relativistic stellar aberration angle for that speed and true stellar angle.

This construction holds for any starship speed and for a star at any angle, including ‘behind’ the starship.

Now that we have aberration in the toolkit, we can go for a ride on a rotating disk.

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