### Usage of $\sim$, $\approx$, $\simeq$, and $\cong$ in observational astronomy?

• My understanding is $\sim$ generally means "on the order of magnitude of" e.g. $T \sim 10^5$ K

$\approx$ is obviously "approximately equal to" so for example one might write $d \approx 400$ pc rather than $d=4 \times 10^2$ pc

$\simeq$ and $\cong$, are where I am more confused as their usage is rarer and less consistent.

The below IAU resolution, for example, seems to use $\simeq$ for truncation $m_{\mbox{bol}} = -26.83199\ldots \simeq -26.832$, but I have also seen $\simeq$ used as "approximately equal to" in other works.

http://arxiv.org/pdf/1510.06262

How should one be using $\sim$, $\approx$, $\simeq$, and $\cong$?

I don't think that there's any set standard for using these approximation signs.

Those are just the pen strokes of hand waving astronomers. :)

To point out, you'd likely receive a very different answer if you posted this on the math stack exchange. I think the distinction between these various symbols has more significance in mathematics. Astronomers on the other hand tend to be less strict and will often just use all of these interchangeably, or you'll see different standards being used by different people. There's less uniformity in their meaning in the sciences than there would be in the mathematics groups.

5 years ago

$\simeq$ and $\approx$ both mean "approximately equal to". I don't think $\cong$ is used so often, but if I read it, I would interpret is as the same as the two others.

$\sim$ in principle means "of the order of", i.e. correct to within an order of magnitude. However, astronomers tend to be rather slobby sometimes (we call all elements except H and He for "metals", we call it "gas" even when it's ionized and should be called "plasma", we omit factors of order unity if we can get away with it, we mix up terms like "flux", "flux density", and "intensity", and so on), and thus $\sim$ is very often used to mean "approximately equal to". It is even sometimes used as "proportional to", but (usually) only when there's no risk of misunderstanding (e.g. $L_\mathrm{UV}\!~\sim\!~\mathrm{SFR}$, meaning that the UV luminosity is proportional to the star formation rate, omitting a factor of $10^{27}$).

Journals usually have rather strict standards for style and nomenclature, so they might not accept a $\sim$ for a $\propto$, or even for a $\simeq$. But if you read a conference proceeding or lecture notes or some other non-refereed text (even a paper on arXiv waiting to be refereed), you best be prepared for anything.