Tools of the Trade

⭐⭐ Dimensional analysis check. For each equation, decide whether it is dimensionally valid; if not, identify what is wrong.

• (a) Orbital velocity $v = \sqrt{GM/r}$.
• (b) Kinetic energy $E = mv$.
• (c) Gravitational potential energy $U = GMm/r^2$.
• (d) Pressure $P = \rho v^2$ ($\rho$ = density).

⭐⭐ Stefan-Boltzmann scaling. $L \propto R^2 T^4$.

• (a) Star A has twice the radius and twice the temperature of Star B; find $L_A/L_B$.
• (b) A red giant has $100\times$ the Sun’s radius but $0.5\times$ its temperature; luminosity in $L_\odot$?
• (c) A white dwarf has $0.01\times$ the Sun’s radius and $4\times$ its temperature; luminosity in $L_\odot$?

⭐ Building physical intuition. Without calculating, predict which way each quantity changes and by roughly what factor:

• (a) orbital period if you double orbital radius (same central mass);
• (b) escape velocity if you double a planet’s mass (same radius);
• (c) luminosity if you double a star’s temperature (same radius);
• (d) dynamical timescale if you double a cloud’s density.

⭐⭐ The dimensions of $G$. Starting from $F = GMm/r^2$:

• (a) derive the dimensions of $G$ in $[L],[M],[T]$;
• (b) verify the CGS unit ($\mathrm{cm^3\,g^{-1}\,s^{-2}}$) matches;
• (c) convert $G = 6.67 \times 10^{-11}~\mathrm{m^3\,kg^{-1}\,s^{-2}}$ to CGS.

⭐⭐ The dimensions of $\sigma$. For $L = 4\pi R^2 \sigma T^4$:

• (a) derive the dimensions of $\sigma$ in $[M],[L],[T],[\Theta]$;
• (b) verify the CGS unit ($\mathrm{erg\,s^{-1}\,cm^{-2}\,K^{-4}}$);
• (c) convert $\sigma = 5.67 \times 10^{-8}~\mathrm{W\,m^{-2}\,K^{-4}}$ to CGS.

⭐⭐ Kepler scaling. Using $P^2 \propto a^3$ (years, AU):

• (a) Neptune orbits at 30 AU — estimate its period;
• (b) an asteroid has $P = 8$ yr — estimate its semi-major axis;
• (c) a Kuiper Belt object at 40 AU — what period?
• (d) Sanity check: Pluto orbits at ~40 AU with period ~248 yr — does your answer agree?

⭐⭐ Fermi estimation: light-minutes. Estimate without a calculator:

• (a) how far light travels in 1 s? in 1 min?
• (b) the Earth–Sun distance is $1~\mathrm{AU} = 1.5 \times 10^{13}$ cm — how many light-minutes is this?
• (c) Jupiter is ~5 AU from the Sun — how many light-minutes?

⭐⭐ Inverse-square law applications.

• (a) The Sun is 1 AU from Earth — how much fainter would it appear from Saturn (10 AU away)?
• (b) Two identical stars have apparent brightnesses in ratio 1
  — ratio of their distances?
• (c) A supernova is visible to the naked eye from 10 kpc — how far could it be observed with a telescope collecting $10{,}000\times$ more light?
• (d) Limiting case: as distance $\to \infty$, what happens to observed flux? What does this imply about detecting objects at arbitrarily large distances?

⭐⭐ Complete workflow: escape velocity. $v_{esc} = \sqrt{2GM/R}$.

• (a) Dimensional check: verify this has dimensions of velocity.
• (b) Ratio method: Earth’s escape velocity is 11.2 km/s — using ratios, estimate the Moon’s ($M_{\mathrm{Moon}} = 0.012\,M_\oplus$, $R_{\mathrm{Moon}} = 0.27\,R_\oplus$).
• (c) Unit conversion: convert Earth’s escape velocity to cm/s.
• (d) OOM check: the Sun’s escape velocity is ~618 km/s — does this make sense given $M_\odot \approx 3 \times 10^5\,M_\oplus$ and $R_\odot \approx 109\,R_\oplus$?

⭐⭐ (Challenge) White dwarf mass-radius relation. Derive why more massive white dwarfs are smaller using dimensional analysis and pressure balance: degeneracy pressure scaling, the $\rho$–$n$ relation, and gravitational pressure. Show the final scaling ($R \propto M^{-1/3}$) and interpret the physics.