Question

The Orion nebula is a nearby ($d sim 500 ext{ pc}$) HII region. It is roughly circular in outline, with an angular radius $ heta = 10^{-3}$ radians. This HII region radiates strong thermal radio continuum emission, where flux density can be described for low frequencies ($ u < 1 ext{ GHz}$) with: $S_ u = 1000 u^2$, (1) where $S_ u$ is in Jy, and $ u$ is given in GHz. This spectrum flattens above $ u = 1 ext{ GHz}$, and at $ u = 23 ext{ GHz}$, $S_ u = 400 ext{ Jy}$. You can assume that the Orion nebula (1) is a circular cylinder whose axis lies along the line-of-sight and whose length equals its diameter, (2) has uniform density and temperature. (a) Estimate the electron temperature of the Orion nebula. (b) Estimate the emission measure ($EM$) at 23 GHz of the nebula, in units of $ ext{cm}^{-6} ext{ pc}$. (c) Using your result from part (b) estimate the electron density in the nebula. Please watch your units!

          The Orion nebula is a nearby ($d sim 500 	ext{ pc}$) HII region. It is roughly circular in outline, with an
angular radius $	heta = 10^{-3}$ radians. This HII region radiates strong thermal radio continuum emission,
where flux density can be described for low frequencies ($
u < 1 	ext{ GHz}$) with:
$S_
u = 1000
u^2$,
(1)
where $S_
u$ is in Jy, and $
u$ is given in GHz. This spectrum flattens above $
u = 1 	ext{ GHz}$, and at
$
u = 23 	ext{ GHz}$, $S_
u = 400 	ext{ Jy}$. You can assume that the Orion nebula (1) is a circular cylinder whose
axis lies along the line-of-sight and whose length equals its diameter, (2) has uniform density and
temperature.
(a) Estimate the electron temperature of the Orion nebula.
(b) Estimate the emission measure ($EM$) at 23 GHz of the nebula, in units of $	ext{cm}^{-6} 	ext{ pc}$.
(c) Using your result from part (b) estimate the electron density in the nebula. Please watch
your units!

The Orion nebula is a nearby (d sim 500 ext pc) HII region. It is roughly circular in outline, with an
angular radius heta = 10^-3 radians. This HII region radiates strong thermal radio continuum emission,
where flux density can be described for low frequencies (u < 1 ext GHz) with:
S
u = 1000
u^2,
(1)
where S
u is in Jy, and u is given in GHz. This spectrum flattens above u = 1 ext GHz, and at
u = 23 ext GHz, S
u = 400 ext Jy. You can assume that the Orion nebula (1) is a circular cylinder whose
axis lies along the line-of-sight and whose length equals its diameter, (2) has uniform density and
temperature.
(a) Estimate the electron temperature of the Orion nebula.
(b) Estimate the emission measure (EM) at 23 GHz of the nebula, in units of extcm^-6 ext pc.
(c) Using your result from part (b) estimate the electron density in the nebula. Please watch
your units!

Added by Josep C.

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