Flux Density - Brightness Temperature Conversion

Start with the Rayleigh-Jeans law:

T = \frac{λ^{2}}{2 k Ω} S

where T is the brightness temperature,

λ

the wavelength, k the Boltzmann constant,

Ω

the beam solid angle, and S the flux density, all in mks units.

Interferometric maps are measured in brightness units

I

of Jy/beam, where “beam” is a nominal area over which the brightness is defined.

We replace

\frac{S}{Ω}

\frac{S}{b e a m} \frac{b e a m}{Ω} = I \frac{b e a m}{Ω}

, where the second term is the conversion of “beam” to a solid angle in units of steradian.

For a "dirty map", the “beam” is the "dirty beam" which has a complicated structure that depends on the number and orientation of baselines as a function of time. Once the image is deconvolved, however, the "dirty beam" is replaced by a "clean beam", a Gaussian withpeak of unity and

θ_{m a j}

and

θ_{m i n}

as the half-power beam widths along the major and minor axes, respectively.

The area of a Gaussian beam is defined by its 2-dimensional integral

Ω = \frac{π θ_{m a j} θ_{m i n}}{4 l n 2}

Substituting

O m e g a

in the first equation and converting all constants to a pre-factor leads to:

T = 0.32 \times 10^{23} \frac{λ^{2}}{θ_{m a j} θ_{m i n}} I

Converting to units to cm, seconds of arc, and mJy/beam results in:

T = 1.36 \frac{λ^{2}}{θ_{m a j} θ_{m i n}} I

or in frequencies

ν

in units of GHz, and I in mJy/beam:

T = 1.222 \times 10^{3} \frac{I}{ν^{2} θ_{m a j} θ_{m i n}}

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