Originally posted by sonhouse
My formula for calculating the first focal point is original, nobody else came up with that.
Taking apart the grav constant and turning it into a number allowed me to rework the equation to give the actual focal point for a given radius and mass. I don't think anyone else even did that, turning the grav constant and c into a number since they are all con ...[text shortened]... ber right, don't have it right in front of me. Or maybe it's 2.9 E-27, have to look up my notes.
When you say the "grav constant" what do you mean? Do you mean Newton's constant G which is 6.674×10^−11 N⋅m^2/kg^2? The dimensions of this constant are metres cubed per second squared per kilogram. If you divide by a couple of factors of the speed of light you will get 7.41x10^-28 metres per kilogram. If you now multiply by a mass you will indeed get a distance, but the distance you should have obtained is called the Schwarzschild radius, apart from a factor of 2.
Wikipedia helpfully gives GM for the sun, it comes out at 1.3271244E-11 km^3/s^2. Dividing this by the speed of light in kilometres per second (3E5 km/s) gets us to 1.474 km, the factor of 2 gets us to 2.948 km, which is the radius of a solar mass black hole. The mass of the Earth is of the order of 300,000 times less so the Schwarzschild radius is of the order of 1 centimetre.
The angle of deflection is 2r/b, where r is the Schwarzschild radius and b the impact parameter (distance of closest approach), for the sun. For an impact parameter of 1 solar radius, in other words for light just grazing the sun, we get 1.75 arc seconds. So we can get the point where parallel rays just grazing the sun converges to as 117,000 solar radii (1/tan(1.75 arc seconds)) and a solar radius is 0.00465 astronomical units to give a focus at 548 astronomical units.
So as long as your formula is something like b^2c^2/GM, where b is the radius of the sun and the other symbols have their usual meanings you will be out by no more than a factor of 4. One case where dimensional analysis (or a hack's guess as it's called in the trade) gives a pretty good answer.