A Mathematical Model For Calculating Surface Brightness

To describe the visible characteristics of extended objects, indicate their overall brilliance and angular dimensions. The overall brilliance, as usual, is given in apparent magnitude. Angular dimensions, according to tradition – in degrees, arcminutes, angular seconds. Denote by m – total brilliance of our extended object, and through A, B – the angular size, respectively the major and minor axes of the ellipse equivalent to the object according to the occupied area of the sky. It is obvious that the larger the number m, A, B, the smaller the apparent brightness of the object’s surface. We say that the average apparent surface brightness, or, simply, the surface brightness of an extended object to the number, calculated by the formula: K = (p2/129600) * (1 / (2,512 m * tg (A / 2) * tg (B / 2))) (1) Unit measurements determined in this way the surface brightness is called Kate. In the formula (1) the number of m, A, B are in the denominator so that they increase the value of K will decrease accordingly. The number m is the exponent of Pogson, reflecting the fact that if m is changed by an amount Dm, then change to a 2,512 Dm times. Tangents at half the angular size of A, B reflect the fact that the telescope to its increase in proportion to increase the tangent of half the angular size g, visible to the naked eye, and with it the celestial object is visible in a telescope is also under increased angle d, the tangent of half of which related to the tangent of g / 2 well-known equation: tg (d / 2) = M * tg (g / 2) , where M is the angular increase the telescope. Therefore, in formula (1) the number of A / 2 and B / 2 are under the tangents, with their increasing value of K will also be proportionally reduced. Coefficient p2/129600 added to the formula (1) exclusively from the anthropic and universal Mathematical considerations in order to value to getting comfortable for a very wide range of surface brightnesses of various celestial objects.

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