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Bonus points for checking your dimensions. Your forumlas omit a lot of constants and assumes no atoms in the ground state, but that does not really matter for getting the units right. For that you need to realize that: The unit for cross section is meter squared [m]x[m], not the reciprocal. When you take e to the power of something, the "something" has to ...

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There are various systems of units that fall within the family colloquially referred to as the metric system. The SI (formerly known by the more descriptive term mks) is based on the meter, kilogram, and second. The cgs system is based on the centimeter, gram, and second. As far as I know, nobody actually uses an mgs (meter-gram-second) system, although it ...

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Among the base units of the International System, the kilogram is the only one whose name and symbol, for historical reasons, include a prefix. Names and symbols for decimal multiples and submultiples of the unit of mass are formed by attaching prefix names to the unit name "gram", and prefix symbols to the unit symbol "g" (CIPM 1967, Recommendation 2). ...

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We are free to define our units however we wish. These are examples where the units are essentially determined by the magnitudes of the constants from certain historically important equations. The simplest example here is the speed of light in vacuum. Its value is defined as $299\,792\,458$ meters per second. Now, the second is already defined with ...

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I am not sure about this, but I think a “measure equation” is something astronomers seem to like a lot: $$\frac\Gamma H \approx \left( \frac T{1.6\cdot 10^{10} \, \mathrm K} \right)^3$$ Or for absolute, relative magnitude and distance (although I am sure I mixed something up): $$m - M = 5 - 5 \log\left(\frac{R}{10 \, \mathrm{pc}}\right)$$ So equations ...

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