Estimating Width of 21cm Line from Atomic Hydrogen Cloud

[v_pino]

Homework Statement

Estimate the width of a 21cm spectral line produced by an interstellar cloud of atomic hydrogen with a temperature of 300K.

Homework Equations

From wiki (link below), I found an equation for full width at half maximum, relating it to Boltzmann constant, temperature T, mass m, and speed of light c.

http://en.wikipedia.org/wiki/Doppler_broadening

I used the mass of one hydrogen atom for m.

The Attempt at a Solution

I got 2.59x10^-4 cm but the answer should be 2.0x10^-4 cm.

Any idea where I went wrong? Is the "width" mentioned in the question equal to the full width at half maximum?

Thanks.

 

[gomboc]

Many astrophysics texts use a less complex approximation for the FWHM line width. Commonly, the derived formula will simply be

$$\Delta\lambda_{FWHM}\approx\frac{2\lambda}{c}\sqrt{\frac{2kT}{m}}$$

If a more in-depth analysis is performed, taking into account the atoms motions in various directions relative to each other and the observers line of sight, you get the more precise formula containing the factor of $$\ln(2)$$ which is missing from the above equation, but present in the one you used.

Your answer is technically more correct, just not the one you were expected to give.

 

[v_pino]

But omitting ln2 gives a larger value of 3.11x10^-6m. The given answer should be 2x10^-6m.

 

[gomboc]

Ah, yes, how stupid of me.

I've had a look through a few textbooks, and I'm not sure what they did to arrive at that solution. The only other thing I could think of would be that rather than using the FWHM line width, they were using the exponential width, i.e. the peak width where intensity drops to $$1/e$$ what it is at the peak, which would give a slightly smaller value. Unfortunately I don't know the formula for finding that offhand. I suppose they could also be assuming some other non-Gaussian velocity distribution (like Maxwellian or something), but that would be unusual for such a question.

 

[paranormal]

I would first like to commend you for using reliable sources and attempting to solve the problem on your own. It shows a strong understanding of research and critical thinking skills.

In regards to your question, the width mentioned in the problem is indeed referring to the full width at half maximum. However, it is important to note that this value is an average and can vary depending on the conditions of the cloud.

In order to obtain a more accurate estimate, it would be beneficial to consider other factors such as the density and velocity of the cloud. These can affect the Doppler broadening and therefore impact the width of the spectral line. Additionally, taking into account the instrumental broadening (due to the resolution of the instrument used to measure the line) can also improve the accuracy of the estimate.

In conclusion, while your attempt at solving the problem is commendable, it is important to consider other factors and sources of error in order to obtain a more precise estimate. Keep up the good work in your scientific endeavors!