Origins of the Elements

Hubble's law $v = H_0 d$ says more distant galaxies recede faster. Running this backwards, all matter was once together, the Big Bang. Space itself is expanding, carrying the galaxies apart.

The CMBR was released about $380{,}000$ years after the Big Bang, when atoms formed and light could travel freely. Cosmic expansion redshifted it into the microwave band at about $2.7\text{ K}$. Its uniform black-body spectrum is strong Big Bang evidence.

$v = H_0 d = (70)(100) = 7000\text{ km/s}$.

Big Bang nucleosynthesis in the first few minutes (at about $10^9\text{ K}$) fused protons and neutrons into helium-4, freezing in the roughly 3:1 hydrogen-to-helium ratio. Stars turn H into He (lowering H), so they cannot produce the observed universal ratio.

Wien's displacement law $\lambda_{\max} T = b$ means $\lambda_{\max} = b/T$: the hotter the star, the shorter its peak wavelength. So a blue star is hotter than a red one.

Each element absorbs its own characteristic wavelengths, leaving a unique pattern of dark lines. Matching that pattern identifies the elements present. (Line shifts give the star's velocity, line broadening gives rotation, and the peak wavelength gives temperature.)

$T = \dfrac{b}{\lambda_{\max}} = \dfrac{2.898\times10^{-3}}{580\times10^{-9}} \approx 5000\text{ K}$ (a yellow, Sun-like star).

Temperature decreases to the right, so "cool" is on the right; luminous is near the top. Cool and luminous is therefore the upper-right, the red giants, whose enormous radius makes them bright despite a low surface temperature.

By Wien's law, hotter stars peak at shorter (bluer) wavelengths. A blue star ($30{,}000$+ K) is far hotter than a red star ($\sim 3000\text{ K}$).

A shift to longer wavelength (redshift) means the source is receding; a shift to shorter wavelength (blueshift) would mean it is approaching. The relation $\dfrac{\Delta\lambda}{\lambda_0} = \dfrac{v}{c}$ then gives the speed.

Main-sequence stars fuse hydrogen into helium (the proton-proton chain in Sun-like stars). About 90% of stars are on the main sequence, where they spend most of their lives.

Luminosity is $L = 4\pi R^2 \sigma T^4$. A red giant is cool (low $T$) but enormous (huge $R$); a white dwarf is very hot (high $T$) but roughly Earth-sized (tiny $R$). Radius and temperature trade off.

$d = \dfrac{v}{H_0} = \dfrac{21{,}000}{70} = 300\text{ Mpc}$.

$\Delta\lambda = 0.9\text{ nm}$, so $v = c\dfrac{\Delta\lambda}{\lambda_0} = (3.0\times10^5)\dfrac{0.9}{500} = (3.0\times10^5)(0.0018) = 540\text{ km/s}$.

$\lambda_{\max} = \dfrac{b}{T} = \dfrac{2.898\times10^{-3}}{10{,}000} = 2.9\times10^{-7}\text{ m} = 290\text{ nm}$, in the ultraviolet/blue. A hot $10{,}000\text{ K}$ star therefore appears blue-white.

Luminosity depends on both radius and temperature, $L = 4\pi R^2 \sigma T^4$. A red giant's radius can be hundreds of times the Sun's, so $R^2$ rises by $\sim 10^4$ while $T^4$ falls only by $\sim 10$, giving a much larger luminosity overall.

At the same temperature $L \propto R^2$, so $\dfrac{L_B}{L_A} = 3^2 = 9$ times as luminous.

$\dfrac{L}{L_\odot} = \left(\dfrac{R}{R_\odot}\right)^2\left(\dfrac{T}{T_\odot}\right)^4 = (0.01)^2(3)^4 = (1\times10^{-4})(81) = 8.1\times10^{-3}$. Even though it is hotter ($T^4$ up $\times 81$), its tiny radius ($R^2$ down $\times 10^4$) makes it far dimmer, only about $0.8\%$ of the Sun's luminosity.

$t \approx \dfrac{1}{H_0} = \dfrac{1}{2.3\times10^{-18}} = 4.35\times10^{17}\text{ s}$. Converting: $\dfrac{4.35\times10^{17}}{3.15\times10^{7}} \approx 1.4\times10^{10}$ years, about 14 billion years (close to the accepted 13.8 billion).

Fusion releases energy only up to iron, the most tightly bound nucleus. Building heavier nuclei absorbs energy, so it happens in the extreme conditions of supernovae (rapid neutron capture), which also blast these elements out into space.

When core hydrogen runs out, the core contracts and heats until helium ignites. Three helium-4 nuclei fuse into carbon-12 (the triple-alpha process), which powers the red-giant phase.

Stellar fusion turns hydrogen into helium (and heavier elements), so over time stars decrease the hydrogen fraction. The observed universal ratio matches Big Bang nucleosynthesis in the first few minutes, when the whole universe was hot and dense enough to fuse helium.

A low-mass star like the Sun swells into a red giant (fusing helium into carbon), then sheds its outer layers, leaving its hot, dense core behind as a white dwarf. Only much more massive stars end as supernovae.

Recession speed first: $v = c\dfrac{\Delta\lambda}{\lambda_0} = (3.0\times10^5)\dfrac{7.5}{500} = 4500\text{ km/s}$. Then Hubble's law: $d = \dfrac{v}{H_0} = \dfrac{4500}{70} \approx 64\text{ Mpc}$. (Stopping at $v$ gives the trap $4500$.)

$L \propto R^2 T^4$, so $\dfrac{L_B}{L_A} = 2^2 \times 3^4 = 4 \times 81 = 324$ times as luminous.