Can you imagine that many hard questions can actually be answered with a few sketches? Do you know that many of your “useless” knowledge can actually help solve big puzzles? In my view, order of magnitude physics should be served as a fundamental course for every physics student. With a few principles, one can miraculously understand within a few minutes those truths that took many great scientists countless efforts to figure out. In fact, one is often advised in research that

 

“Never calculate without first knowing the answer.” – John A. Wheeler

 

To get a taste of this fantastic topic, you can go through Guesstimation and Guesstimation 2.0. A more advanced lecture note on order of magnitude physics can be found here, or a simplified version here. If you need some references for constants, check the PDG document for physical and astrophysical constants.

In the following, I list some important scales in physics and astrophysics for future reference. Regarding mathematical symbols for equivalence, I will follow the conventions:

  • “$a\approx b$” means that $a$ and $b$ are equal up to the 4/5 rounding rule for the last digit. Examples: $3.14\approx 3$, $1.55\approx 1.6$.
  • “$a\sim b$” means that $a$ and $b$ are equal within an order of magnitude, i.e. a factor of $10$, while I will try to minimize the error within a factor of $3$. For example, both $4\times 10^3 \mathrm{m} \sim 10^3\mathrm{m}$ and $4\times 10^3 \mathrm{m} \sim 10^4\mathrm{m}$ are correct, but I prefer the latter. Also note that if I write something like $a\sim b\sim c\sim d$, I mean that all $a,b,c,d$ are equal within an order of magnitude – the error should not be amplified.
  • “$a\simeq b$” means that $a$ and $b$ are equal up to a factor of $\sqrt{10}\approx 3.16$.
  • “$a\propto b$” means that $a$ is proportional to $b$, while the value of them may differ a lot or even do not have the same dimension. For example, time=distance/speed, we may write $t\propto s$.

Order of magnitudes

  • Units:
    • $\hbar c \simeq 0.2 ~\mathrm{GeV ~fm}$.
    • $1~\mathrm{yr} \simeq \pi\times 10^7 ~\mathrm{s}$.
    • $1~\mathrm{pc} \simeq 3~\mathrm{lyr} \simeq 3\times 10^{16}\mathrm{m}$.
  • Particles:
    • $m_\mathrm{proton} \simeq m_\mathrm{neutron} \simeq 2000~m_\mathrm{electron} \simeq 1 ~\mathrm{GeV} \simeq 10^{-27} \mathrm{kg}$.
    • $r_\mathrm{c,proton} = \frac{2\pi}{m_\mathrm{proton}} \simeq 1~\mathrm{fm}$.
  • Stars and planets:
    • $M_\odot \simeq 10^3 M_\mathrm{jupiter} \simeq 3\times 10^5 M_\oplus \simeq 3\times 10^7 M_\mathrm{moon} \simeq 10^{57} \mathrm{GeV}$.
    • $R_\odot/c \simeq 100~R_\oplus/c \simeq 2 ~\mathrm{s}$.
    • $R_\mathrm{s,\odot} = \frac{2GM_\odot}{c^2} \simeq 3 ~\mathrm{km}$; $R_\mathrm{s,\oplus} = \frac{2GM_\oplus}{c^2} \simeq 1 ~\mathrm{cm}$.
    • Distance between the sun and the earth is $\simeq$ 8 light minutes; distance between the earth and the moon is $\simeq$ 1 light second.
  • Compact objects:
    • $M_\mathrm{M87}\sim 10^9 M_\odot$; $M_\mathrm{MilkyWay}\sim 10^6M_\odot$.
  • Milky Way galaxy:
    • $M_\mathrm{Milky Way}\sim 10^{12}M_\odot$; $N_\mathrm{stars}\sim 10^{11}$
    • Distance between the sun and galaxy center $r_\odot \sim 8~\mathrm{kpc}$; NFW profile scale radius $r_s\sim 20 ~\mathrm{kpc}$.
  • The observable universe:
    • Critical density $\rho_\mathrm{crit} \simeq 10^{-29}\mathrm{g/cm^3} \simeq 5~\mathrm{protons/m^3} \simeq 10^{11} M_\odot/\mathrm{Mpc^{-3}}$
    • Hubble parameter $H_0\simeq 10^{-33}~\mathrm{eV}$, $H_0^{-1} \simeq 10^{10} \mathrm{yr} \simeq 10^{10} \mathrm{ly} \simeq 5000 \mathrm{Mpc}$.
    • Matter-radiation equality $k_\mathrm{eq}\simeq 0.01 \mathrm{Mpc^{-1}}$, $H_{eq}\simeq 10^{-28} \mathrm{eV}$.
  • Dark matter
    • $\rho_c \simeq 1 ~\mathrm{GeV/m^3}\simeq 8\times 10^{-48} ~\mathrm{GeV^4}$.
    • $\rho_\mathrm{local} \simeq 0.4 ~\mathrm{GeV/cm^3} \simeq 3\times 10^{-42} \mathrm{GeV^4}$.

Conversion table for natural units

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Electromagnetic spectrum

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US units

The units in US are a little bit confusing (at least for me for a long time).

  • Length: $1~\mathrm{mile}\approx 1.6~\mathrm{km}$; $1~\mathrm{feet} \approx 0.3~\mathrm{m}$; $1~\mathrm{inch} = 2.54~\mathrm{cm}$.
  • Volume: $1~\mathrm{gal} \approx 3.8~\mathrm{L}$; $1~\mathrm{fl~oz} \approx 30~\mathrm{ml}$.
  • Weight: $1~\mathrm{lb} \approx 0.45~\mathrm{kg}$; $1~\mathrm{oz}\approx 28~\mathrm{g}$.
  • Temperature: $[\mathrm{^\circ C}] = \frac{5}{9}([\mathrm{^\circ F}]-32)$. $32~\mathrm{^\circ F} = 0~\mathrm{^\circ C}$; $77~\mathrm{^\circ F} = 25~\mathrm{^\circ C}$; $212~\mathrm{^\circ F} = 100~\mathrm{^\circ C}$.