# Optical Thickness of the Atmosphere

Homework question:

Show that the absorption optical thickness of the atmosphere at pressure $$p$$ (i.e., between $$p$$ and the top of the atmosphere) as a function of the mixing ratio of the absorbing gas $$r$$ and the absorption coefficient per unit mass $${k}_{ν}$$ can be expressed as:

$$τ = \frac{k_νrp}{g}$$.

Assume $$r ≪ 1$$

(So we can approximate $$r ≃ \frac{ρ_a}{ρ}$$, where $$ρ_a$$ is the density of the absorbing gas), and that $$r$$ and $$k_ν$$ are vertically uniform, and assume that there is no scattering.

Has the anthropogenic increase in $$CO_2$$ shifted the emission level to higher or lower altitude? Prove your answer.

Attempt:

Given the equation $$τ = \frac{k_νrp}{g}$$

I took the $$ln(\tau) = ln(\frac{k_νrp}{g})$$

$$ln(\tau) = ln(k_ν) + ln(r) + ln(p) - ln(g)$$

$$dln(\tau) = dln(k_ν) + dln(r) + dln(p) - dln(g)$$

$$\frac{dln\tau}{\tau} = \frac{dlnk_ν}{k_v} + \frac{dlnr}{r} + \frac{dlnp}{p} - \frac{dlng}{g}$$

We will assume that $$g$$ and $$k_v$$ are constant, so disregard for a moment here. It is a battle between the mixing ratio $$r$$ and $$p$$.

$$\frac{dln\tau}{\tau} = \frac{dlnr}{r} + \frac{dlnp}{p}$$

If we want to know if peak $$\tau$$ moves up or down, I say that $$\tau$$ is constant. If I increase the mixing ratio $$r$$ (increase $$CO_2$$ concentration in atmosphere), pressure must decrease in order to keep $$\tau$$ constant, and decreased pressure means moving up in altitude. I conclude that increasing $$CO_2$$ concentrations shifted the emission level to a higher altitude.

Help

I am unsure how to arrive at the initial given equation. My understanding of $$\tau$$ is minimal, and I just see too many variations and integrals concerning $$\tau$$. I see how the Ideal Gas Law and the Hypsometric Relation will be used to replace variables

$$ρ = ρ_o*e^\frac{-z}{H}$$ where $$H = \frac{RT}{g}$$

$$P = ρRT$$

(just trying to work backwards and piece the puzzle together, but the vision is unclear), but what was the original equation that was used, and ultimately--why?

• Hi PattyWatty, and welcome to SE. This is not primarily a homework help site, and in general if you just copy/paste a question in here people will not answer it. However, if you need help with a specific concept, that's a different matter. Please have a read of earthscience.meta.stackexchange.com/a/317/39 . – Semidiurnal Simon Sep 29 '18 at 20:17
• I'm voting to put this question on hold as off-topic because it is a homework question with no attempt at an answer. The asker can edit it to be more acceptable if they wish. – Semidiurnal Simon Sep 29 '18 at 20:18
• @SemidiurnalSimon Thank you for the advice, I do need help with specific content. I will edit accordingly. Please, do not delete/down-vote, the atmosphere is my passion, this is the only community I know of that I can create a dialogue with. – PattyWatty27 Sep 29 '18 at 20:34
• Not my down-votes :-) But don't worry - standard procedure here is to "vote to close", which means (once a few more reviewers see it) it gets put "on hold" to give you time to edit the question. If you edit a question that's on hold, it automatically gets sent to reviewers to see if it should be re-opened. – Semidiurnal Simon Sep 29 '18 at 20:39