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I was reading the abstract of a paper regarding attempts at detection of a molecular precursor to the amino acid glycine, and I came across concepts (and notation) that I found confusing. If someone could describe in layman's terms (I've taken an astrophysics class so I'm not totally unaware) that would be great.

In the question I refer to the following section of a paper's abstract:

"we derive upper limit column densities of NH$_2$OH of $\leq 1.4 \times 10^{13} \mathrm{cm}^{-2}$ and $\leq 1.5 \times 10^{13} \mathrm{cm}^{-2}$ toward the B1 and B2 shocks, respectively, and upper limit relative abundances of $N_\mathrm{NH_2OH}/N_\mathrm{H_2}$ $\leq 1.4 \times 10^{-8}$ and $\leq 1.5 \times 10^{-8}$, respectively."

The paper in question is called CSO and CARMA Observations of L1157. I. A Deep Search for Hydroxylamine (NH$_2$OH), and located here.

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A column density is just the integral of a (number) density along the line of sight.

Often, you would get the same observational absorption or emission signature from a number density of $n$ molecules cm$^{-3}$, spread along a line of sight through a cloud of diameter $x$, as you would with a density of $0.1n$ spread over $10x$.

In other words, all we can infer is the product or, more accurately, the integral of $n\ dx$, and this is known as the column density.

In the paper you refer to, I doubt they know the thickness of the shock they are investigating, so a column density is all that can be determined.

The relative abundance is more straighforward; it is just the ratio of the number densities of the two labelled species.

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