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Homework 3

ASTR670

Spring 2009Prof. Alberto Bolatto

1) Look at Figure 1 in page 223 of the Dickey & Lockman [1990] review. Itrepresents the line profiles due to a combination of cold and warm HI alongthe line of sight.a. Using the cloud properties there listed, calculate the precise observed

brightness temperature line profiles to reproduce both bottom panels ofthat figure. Consider both situations, with the warm gas in front andbehind the cold gas. Assume that the intrinsic line profiles are Gaussian,with a=Av/2.35 [Av is usually quoted as the full-width at half-maximumof a Gaussian in radioastronomy].

b. What would be the sizes of these clouds, if they were spherical andvirialized?

c. What would be the column density of the warm cloud to obtain the samebrightness temperature, if its kinetic temperature was 7,000 K?

2] What is wrong with this paper? Read Grimm et al. [2007], ApJS, 173, 70 andfind the error that went past the authors and the referee. With yourknowledge of the multiphase ISM,what would you do to improve Figure 6 ofthat paper? [Hint: it has something to do with a misused dataset]

3] The equation for the Einstein A coefficient for rotational dielectric dipoletransitions is

64jtV 2JU

where Ju is the rotational quantum number of the upper level, gu is thecorresponding statistical weight, and [i is the electric dipole.a. Assuming that the excitation temperature Tex is constant along the line of

sight, show that the equation for the opacity xv as a function of the columndensity of the upper level is

/ hv \ekT"-\

8tt3 J„3/z ** gu

\ /

where AV is the velocity line-width of the transition.b. Compute the ratio of brightness temperature for two consecutive

transitions in the optically thin limit, and plot it as a function of excitationtemperature.

c. What is this ratio in the optically thick limit?

AV

Homework 3

ASTR670

Spring 2009Prof. Alberto Bolatto

1] Look at Figure 1 in page 223 of the Dickey & Lockman [1990] review. Itrepresents the line profiles due to a combination of cold and warm HI alongthe line of sight.a. Using the cloud properties there listed, calculate the precise observed

brightness temperature line profiles to reproduce both bottom panels ofthat figure. Consider both situations, with the warm gas in front andbehind the cold gas. Assume that the intrinsic line profiles are Gaussian,with g=Av/2.35 [Av is usually quoted as the full-width at half-maximumof a Gaussian in radioastronomy].

b. What would be the sizes of these clouds, if they were spherical andvirialized?

c. What would be the column density of the warm cloud to obtain the samebrightness temperature, if its kinetic temperature was 7,000 K?

2] What is wrong with this paper? Read Grimm et al. [2007], ApJS, 173, 70 andfind the error that went past the authors and the referee. With yourknowledge of the multiphase ISM, what would you do to improve Figure 6 ofthat paper? (Hint: it has something to do with a misused dataset]

3] The equation for the Einstein A coefficient for rotational dielectric dipoletransitions is

_64*V ,i"' 3/zc3 Mgu

where ju is the rotational quantum number of the upper level, gu is thecorresponding statistical weight, and \i is the electric dipole.a. Assuming that the excitation temperature Tcx is constant along the line of

sight, show that the equation for the opacity xv as a function of the columndensity of the upper level is

8jt3 2J.. — AN.T.. = U

1 hv

-1

\ /3/i g„k JAV

where AV is the velocity line-width of the transition.b. Compute the ratio of brightness temperature for two consecutive

transitions in the optically thin limit, and plot it as a function of excitationtemperature.

c. What is this ratio in the optically thick limit?