Monday, November 8, 2010

Basic Problems in Astrophysics (5)

The Rosette Nebula: The basis for the Stromgren Sphere problem shown.

We look now at another basic problem associated with what is called "the Stromgren sphere", which is defined as the effective range or radius of radiative influence of an exciting source- which depends upon the type of star (or other emission object, e.g. emission nebula). The radiative intensity of the source then determines the dimensions of the Stromgren sphere, and other associated parameters.

In this example illustration, we consider the Rosette Nebula (shown) which is taken to have a brightness temperature of 200K, measured at 242 MHz, and assuming an electron temperature of 10,000K. We then need to find:

a)the optical depth (assuming optically thin conditions)

b)the emission measure

c)the average electron density assuming the radius of the Stromgren sphere is 37 pc


For an electron temperature T(e) and a brightness temperature, T(b) , the optical depth (t_d) can be obtained from:

T(b) = T(e) [1 – exp(-t_d)]

Where t denotes the optical thickness. For the optically thin approximation, t_d << 1 so we can write:

T(b) = T(e)(t_d) So: t_d = T(b)/ T(e) = 200K / 10,000 K = 0.02

(b) The optical depth as a function of the emission measure (EM) can be written:

t_d = 0.4/f^2 INT _0 to L {N^2 ds } = 0.4/f^2 (EM)

where f is the frequency in megahertz (Mhz), INT denotes integral (from 0 to L), N is the electron number density and L is the path length in parsecs. Writing the emission measure as a function of optical depth, we get (noting the units are customarily expressed in electrons per cm^3 for N and parsecs for r, with f in MHz)

EM = t_d f^2/ 0.4 = 2.5 (t_d)(f^2)= (0.02) (242)^2 (2.5) = 2.9 x 10^3

(c) The number density is easily obtained from EM using the integral from (b), since:

INT _0 to L {N^2 ds } = (EM)

Integrating from 0 to L over path ds:

N^2 L = EM

And N = SQRT[EM/L] = SQRT [2.9 x 10^3/ 37] = 8.89

or about 9 electrons per cubic centimeter. Note that we use 37 pc for the path length, i.e. effectively the diameter of the Stromgren sphere for the Rosette Nebula.

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