# Why are these equations valid despite seemingly inconsistent units?

I am having quite a difficult time in trying to understand what units are used in this paper and how to convert things to SI. For example, look at equation (1): $$T_M \approx 1500 \rho^{1/3}\ \mathrm{K}\tag{1}$$ It seems to be showing that temperature is measured in units of $\mathrm{g\,cm^{-3}\,K}$. Then look at equation (2), $$T_M \approx 2800 \rho^2\ \mathrm{K}\tag{2}$$ which seems to be showing that temperature is measured in $\mathrm{g^2\,cm^{-6}\,K}$. Equation (10) doesn't make sense with these either: $$\sigma \approx \frac{5\times 10^{20}\rho^{4/3}}{T(1 + 3x)}\mathrm{esu}\tag{10}$$ How are these consistent?

• Jan 22, 2016 at 4:06
• Yes, that's bad style, even though the commentary "where $\rho$ is the density in $gcm^{-3}$" makes it right. What it means is that the authors expect the reader to strip the units from the value of the density and insert it into the equation as a raw number. They should have written $T_M\approx 1500\rho^{1/3}Kcmg^{-1/3}$, instead. Admittedly, that looks very "funny" in itself. Jan 22, 2016 at 5:15
• One sees this too often. Another way of looking at it is that the author neglected to ascribe units to the prefactor: 1500 K cm g${}^{-1/3}$ Jan 22, 2016 at 14:20

Each equation contain a different arbitrary constant: 1500, 2800, and 5 E20.

It can be assumed that each arbitrary constant has exactly the right units to make everything come out right...

It is sloppy to not specify the units of these constants...

Edited for example:

I could conduct experiments on the dynamics of falling objects, and publish that the distance of fall from rest, in metres (D), and the time of fall, in seconds, (t) seem to be related, and the best fit gives:$$D \approx4.9 \times t^2$$without implying that the units of distance are time squared...

• I agree. It's more than sloppy. Unclear equations can lead to misinterpretations, mistakes and disaster. The difference between a stable orbit and a satellite crashing into a planet. Jan 22, 2016 at 15:02
• Even under this interpretation, it's very odd that equation (10) includes explicit units at the end, which are not the units of the resulting quantity. Jan 22, 2016 at 16:39

The units are not consistent. Or in less precise terms, wrong.

Here's the only way I can think of for this to make some sense: just after equation (1), the paper says

...where $\rho$ is the density in $\mathrm{g\,cm^{-3}}$.

My guess is that they intend you to take $\rho$ as a pure number. For example, if the density is $0.1\ \mathrm{g\,cm^{-3}}$, then you should take $\rho = 0.1$. But that's inconsistent where the part just above equation (2) where it says

For $\rho \gtrsim 0.4\ \mathrm{g\,cm^{-3}}$...

which requires that $\rho$ actually have units in it.

I suppose the intent is that you always consider $\rho$ to be either $\text{density}$ or $\frac{\text{density}}{\mathrm{g\,cm^{-3}}}$, as needed. One would hope that, especially in modern times, this sort of sloppiness with units becomes increasingly rare, because it is confusing.

• Ok. We can treat the density to be a pure number if we want to calculate. but what of equation 10? But how does this fit into making sense of equation 10? Jan 22, 2016 at 12:50
• I've edited accordingly. Jan 22, 2016 at 14:09
• Thank you. But doing so would leave a number in units of esu (electrostatic units), which I understand to be equivalent to charge or statC which can be expressed alternatively as $cm^{3/2}g^{1/2}s^{-1}$. But apparently the electrical conductivity in cgs units can be given by 1/s or s^-1. How does this fit into our understanding? Jan 22, 2016 at 15:31
• Oh, right, I wasn't paying attention. Nah, I can't make any sense of that one. Jan 22, 2016 at 16:31
• Though Salpeter was a perspicacious physicist, both of us graduate students cannot make sense of this. It's quite frustrating, for I want to try to incorporate the paper's electrical and thermal conductivities, if only I could make sense of these limiting conventions. Jan 22, 2016 at 16:34