Part I: Stellar Physics | Chapter 3

Stellar Evolution

From the main sequence to stellar death: main-sequence lifetimes, red giant evolution, the Chandrasekhar mass, core collapse, and mass loss mechanisms

Overview

Stellar evolution is the process by which a star changes over time as it exhausts its nuclear fuel. The life story of a star is determined primarily by its initial mass: low-mass stars ($M \lesssim 8 M_\odot$) end as white dwarfs, while massive stars ($M \gtrsim 8 M_\odot$) undergo core collapse to form neutron stars or black holes. This chapter traces the evolutionary pathways from the main sequence through the red giant phase to the final stellar remnants.

The theoretical framework for stellar evolution was largely developed in the 1950s and 1960s through pioneering computational work by Schwarzschild, Hayashi, Henyey, and others. Modern stellar evolution codes can track the life of a star from birth to death with remarkable precision.

1. Main Sequence Lifetime

The main sequence is the longest phase in a star's life, during which it burns hydrogen in its core. The lifetime on the main sequence is determined by the available nuclear fuel and the rate at which it is consumed (the luminosity).

1.1 Nuclear Energy Budget

The total nuclear energy available from hydrogen burning is:

$$E_{\text{nuc}} = \eta M_{\text{fuel}} c^2 = \eta f M c^2$$

where $\eta = 0.007$ is the mass-to-energy conversion efficiency for hydrogen fusion ($4p \to \!^4\text{He}$ converts 0.7% of the rest mass to energy),$f \approx 0.10{-}0.13$ is the fraction of the total stellar mass that participates in core hydrogen burning, and $M$ is the stellar mass.

1.2 Derivation of the Main Sequence Lifetime

The main-sequence lifetime is:

$$t_{\text{MS}} = \frac{E_{\text{nuc}}}{L} = \frac{\eta f M c^2}{L}$$

Using the mass-luminosity relation for main-sequence stars, $L \propto M^{3.5}$(derived from homology arguments in the next chapter), we can express the luminosity as:

$$L = L_\odot \left(\frac{M}{M_\odot}\right)^{3.5}$$

Substituting:

$$t_{\text{MS}} = \frac{\eta f M c^2}{L_\odot (M/M_\odot)^{3.5}} = \frac{\eta f M_\odot c^2}{L_\odot} \left(\frac{M}{M_\odot}\right)^{-2.5}$$

Evaluating the prefactor with $\eta = 0.007$, $f = 0.10$:

$$\frac{\eta f M_\odot c^2}{L_\odot} = \frac{0.007 \times 0.10 \times (2 \times 10^{30})(3 \times 10^8)^2}{3.8 \times 10^{26}} \approx 3.3 \times 10^{17} \text{ s} \approx 10^{10} \text{ yr}$$

Main Sequence Lifetime

$$\boxed{t_{\text{MS}} \approx 10^{10} \left(\frac{M}{M_\odot}\right)^{-2.5} \text{ years}}$$

The Sun's main-sequence lifetime is about 10 billion years. A $10 M_\odot$ star lives only $\sim 30$ million years, while a $0.1 M_\odot$ red dwarf can burn for $\sim 10^{13}$ years — far longer than the current age of the universe.

1.3 The Main Sequence as a Mass Sequence

The position of a star on the main sequence is determined almost entirely by its mass. More massive stars are hotter, more luminous, and larger:

$$L \propto M^{3.5}, \qquad R \propto M^{0.7}, \qquad T_{\text{eff}} \propto M^{0.55}$$

These scaling relations follow from the stellar structure equations under the homology assumption (that all main-sequence stars have the same dimensionless structure).

2. Red Giant Evolution

When hydrogen is exhausted in the core, the star undergoes dramatic structural changes that transform it from a compact main-sequence star into an enormous red giant.

2.1 Hydrogen Shell Burning

After core hydrogen exhaustion, the inert helium core contracts on a Kelvin-Helmholtz timescale. As the core contracts, gravitational energy is released, heating the hydrogen-rich material surrounding the core. A hydrogen-burning shell ignites at the core-envelope boundary.

The shell-burning luminosity is much higher than the core-burning luminosity was, because the shell has a larger surface area and operates at a higher temperature (the contracting core has heated it). The luminosity increases and the star moves off the main sequence.

2.2 The Mirror Principle

A fundamental principle of stellar evolution states that the core and envelope respond in opposite ways to changes in nuclear burning:

The Mirror Principle

When a burning shell is active, the core and envelope behave as “mirrors”:

If the core contracts, the envelope expands
If the core expands, the envelope contracts

This can be understood from the virial theorem: as the core contracts, it releases gravitational energy. Part of this energy goes into the shell luminosity, and part into expanding the envelope. The expanding envelope cools, lowering the surface temperature and making the star red.

The physical mechanism underlying the mirror principle involves the response of the burning shell to changes in the core mass. As the shell burns, it adds mass to the core, which contracts under its increased weight. The contraction releases gravitational energy given by:

$$\Delta E_{\text{grav}} \sim \frac{G M_c \Delta M_c}{R_c}$$

where $M_c$ and $R_c$ are the core mass and radius. By the virial theorem, half of this energy goes into thermal energy of the core (heating it further) and half is radiated or used to expand the envelope.

2.3 The Red Giant Branch

As the helium core grows in mass and contracts, the envelope expands enormously. For a solar-mass star, the radius increases from $\sim 1 R_\odot$ to $\sim 200 R_\odot$, while the surface temperature drops from $\sim 5800$ K to $\sim 3500$ K. The luminosity increases to $\sim 2000 L_\odot$ at the tip of the red giant branch.

The red giant branch terminates when the helium core reaches a critical temperature ($\sim 10^8$ K) for helium ignition. In low-mass stars ($M \lesssim 2 M_\odot$), the core becomes electron-degenerate before helium ignites, leading to the helium flash — a thermonuclear runaway in degenerate matter.

2.4 Subsequent Evolution

After helium ignition, the star settles onto the horizontal branch (or red clump), burning helium in its core and hydrogen in a shell. When core helium is exhausted, the star ascends the asymptotic giant branch (AGB) with double shell burning (He and H shells). AGB stars experience thermal pulses — periodic helium shell flashes that dredge up nucleosynthesis products to the surface.

3. The Chandrasekhar Mass

Subrahmanyan Chandrasekhar showed in 1930 that there is a maximum mass for a white dwarf supported by electron degeneracy pressure. This limit fundamentally determines which stars end as white dwarfs and which undergo core collapse.

3.1 Electron Degeneracy Pressure

At high densities, the Pauli exclusion principle forces electrons into higher and higher momentum states. The resulting degeneracy pressure is a quantum mechanical effect independent of temperature.

In the non-relativistic limit, the Fermi momentum is determined by the electron number density $n_e$:

$$p_F = \hbar(3\pi^2 n_e)^{1/3}$$

The electron number density is related to the mass density through $n_e = \rho/(\mu_e m_H)$, where $\mu_e$ is the mean molecular weight per electron ($\mu_e = 2$ for a fully ionized He/C/O composition). The non-relativistic degeneracy pressure is:

$$P_{\text{NR}} = \frac{(3\pi^2)^{2/3} \hbar^2}{5 m_e} n_e^{5/3} = K_{\text{NR}} \rho^{5/3}$$

This is a polytrope with $\gamma = 5/3$ (polytropic index $n = 3/2$). In the ultra-relativistic limit ($p_F \gg m_e c$):

$$P_{\text{UR}} = \frac{(3\pi^2)^{1/3} \hbar c}{4} n_e^{4/3} = K_{\text{UR}} \rho^{4/3}$$

This is a $\gamma = 4/3$ polytrope (index $n = 3$).

3.2 Derivation of the Chandrasekhar Limit

For the non-relativistic case ($n = 3/2$ polytrope), the mass-radius relation from the Lane-Emden equation gives:

$$M \propto R^{-3} \rho_c \propto R^{-3} (M/R^3)$$

More precisely, using the Lane-Emden solution:

$$R \propto M^{-1/3}$$

As mass increases, the radius decreases, increasing the central density and the Fermi momentum. When $p_F \sim m_e c$, the electrons become relativistic and the equation of state softens to $\gamma = 4/3$.

The critical insight is that for $\gamma = 4/3$ (the $n = 3$ polytrope), the mass becomes independent of central density (and hence of radius). This unique mass is the Chandrasekhar mass. From the Lane-Emden equation for $n = 3$:

$$M_{\text{Ch}} = 4\pi \left(\frac{K_{\text{UR}}}{G}\right)^{3/2} \frac{1}{\pi} \left(-\xi_1^2 \theta'(\xi_1)\right)_{n=3}$$

Substituting the Lane-Emden values ($\xi_1 = 6.897$, $-\xi_1^2\theta'(\xi_1) = 2.018$) and the constants:

Chandrasekhar Mass

$$\boxed{M_{\text{Ch}} = \frac{5.83}{\mu_e^2} M_\odot \approx 1.44 M_\odot}$$

for $\mu_e = 2$ (carbon/oxygen white dwarf). This can be written more explicitly as:

$$M_{\text{Ch}} = \frac{\omega_3}{\sqrt{\pi}} \left(\frac{\hbar c}{G}\right)^{3/2} \frac{1}{(\mu_e m_H)^2}$$

where $\omega_3 \approx 2.018$ is the Lane-Emden eigenvalue.

3.3 Physical Meaning

The Chandrasekhar mass can be understood from a simple dimensional argument. The only mass scale that can be formed from $\hbar$, $c$, $G$, and $m_H$ is:

$$M_{\text{Ch}} \sim \left(\frac{\hbar c}{G m_H^2}\right)^{3/2} m_H \sim \left(\frac{m_{\text{Pl}}}{m_H}\right)^3 m_H$$

where $m_{\text{Pl}} = \sqrt{\hbar c/G}$ is the Planck mass. The ratio$m_{\text{Pl}}/m_H \sim 10^{19}$ gives $M_{\text{Ch}} \sim 10^{57} m_H \sim 2 M_\odot$, remarkably close to the exact value.

Stars with initial mass $M \lesssim 8 M_\odot$ lose enough mass during the AGB phase to leave a white dwarf with $M_{\text{WD}} < M_{\text{Ch}}$. More massive stars retain cores exceeding the Chandrasekhar limit, and electron degeneracy pressure cannot halt the collapse.

4. Core Collapse in Massive Stars

Stars with initial mass $M \gtrsim 8 M_\odot$ burn through successive nuclear fuels — H, He, C, Ne, O, Si — building up an iron core. When the core mass exceeds the Chandrasekhar limit, catastrophic collapse ensues.

4.1 Onion-Shell Structure

Before collapse, a massive star has a layered structure like an onion, with progressively heavier elements burning in concentric shells:

Fe core → Si/S shell → O/Ne shell → C/O shell → He shell → H envelope

Each successive burning stage is shorter: H burning $\sim 10^7$ yr, He $\sim 10^6$ yr, C $\sim 10^3$ yr, O $\sim$ months, Si $\sim$ days.

4.2 Iron Photodisintegration

Iron-56 has the highest binding energy per nucleon, so no further energy can be extracted by fusion. When the core temperature exceeds $\sim 5 \times 10^9$ K, photons are energetic enough to photodisintegrate iron:

$$\!^{56}\text{Fe} + \gamma \to 13\,\!^4\text{He} + 4n \quad (Q = -124.4 \text{ MeV})$$

$$\!^4\text{He} + \gamma \to 2p + 2n \quad (Q = -28.3 \text{ MeV})$$

This is endothermic — it absorbs energy rather than releasing it. The energy absorbed per nucleon is about 8.8 MeV, which removes the thermal pressure support that was maintaining the core against gravity. The photodisintegration effectively undoes all the nuclear burning that took millions of years, in a fraction of a second.

4.3 Electron Capture and Neutronization

Simultaneously, the high densities ($\rho \gtrsim 10^{10}$ g/cm³) drive electron capture onto protons and nuclei:

$$e^- + p \to n + \nu_e$$

This removes electrons, reducing the electron degeneracy pressure that was supporting the core. The electron fraction drops from $Y_e \approx 0.42$ to $\sim 0.35$. Each capture also produces a neutrino that initially escapes, carrying away energy and further accelerating the collapse.

4.4 Collapse and Bounce

With both thermal pressure (photodisintegration) and degeneracy pressure (electron capture) removed, the iron core collapses in near-free-fall:

$$t_{\text{collapse}} \sim t_{\text{ff}} = \sqrt{\frac{3\pi}{32 G \rho}} \sim 0.1 \text{ s}$$

The inner core ($\sim 0.6 M_\odot$) collapses homologously (maintaining a self-similar velocity profile $v \propto r$), while the outer core falls supersonically.

At nuclear density ($\rho \sim 2.7 \times 10^{14}$ g/cm³), the strong nuclear force becomes repulsive and the equation of state stiffens dramatically. The inner core bounces, sending a shock wave outward through the infalling material. The shock initially stalls at $\sim 100{-}200$ km as it loses energy to iron photodisintegration and neutrino losses.

Core Collapse Energy Budget

Gravitational energy released: $\sim 3 \times 10^{53}$ erg

Neutrino energy: $\sim 3 \times 10^{53}$ erg (99%)

Kinetic energy of ejecta: $\sim 10^{51}$ erg (1%)

Radiated light: $\sim 10^{49}$ erg (0.01%)

The supernova explosion carries only $\sim 1\%$ of the total gravitational energy. The vast majority escapes as neutrinos over $\sim 10$ seconds.

The mechanism by which the stalled shock is revived to produce a successful supernova explosion remains one of the great unsolved problems in astrophysics. The leading mechanism is neutrino-driven convection: a small fraction ($\sim 5\%$) of the outgoing neutrino luminosity is reabsorbed behind the shock, heating the material and reviving the explosion.

5. Stellar Mass Loss

Stars lose mass throughout their lives through various mechanisms. Mass loss profoundly affects stellar evolution, determining the final remnant mass and enriching the interstellar medium with processed material.

5.1 Radiation-Driven Winds

Hot, luminous stars drive powerful winds through radiation pressure on spectral lines. Photons are absorbed and re-emitted by ions in the stellar atmosphere, transferring momentum to the gas. The mass-loss rate can be estimated from the momentum budget:

$$\dot{M} v_\infty \sim \frac{L}{c}$$

where $v_\infty$ is the terminal wind velocity. For O-type stars,$v_\infty \sim 2000{-}3000$ km/s and $\dot{M} \sim 10^{-6} M_\odot$/yr. The actual momentum transfer can exceed $L/c$ due to multiple scattering (the same photon can be absorbed and re-emitted multiple times in different spectral lines).

The Castor-Abbott-Klein (CAK) theory gives the mass-loss rate as:

$$\dot{M}_{\text{CAK}} \propto L^{1/\alpha'} M^{1-1/\alpha'}$$

where $\alpha' \approx 0.6$ is the CAK force multiplier parameter.

5.2 AGB Mass Loss and Planetary Nebulae

Asymptotic giant branch stars lose mass at rates up to $10^{-4} M_\odot$/yr through a combination of pulsations and radiation pressure on dust grains. Pulsations (Mira variables) levitate gas to radii where temperatures are cool enough ($\sim 1500$ K) for dust to condense. Radiation pressure then accelerates the dust, which drags the gas along by collisions.

A superwind phase at the tip of the AGB ejects most of the remaining envelope in $\sim 10^4$ years, creating a planetary nebula when the exposed hot core ionizes the ejected material. The remaining core becomes a white dwarf.

5.3 Wolf-Rayet Stars

Wolf-Rayet (WR) stars are the stripped cores of initially very massive stars ($M_{\text{init}} \gtrsim 25 M_\odot$) that have lost their hydrogen envelopes through powerful winds. They show strong emission lines of He, C, N, or O and have mass-loss rates of $\dot{M} \sim 10^{-5} M_\odot$/yr with wind velocities $\sim 2000$ km/s.

The cumulative mass loss determines the initial-to-final mass relation:

$$M_{\text{final}} = M_{\text{initial}} - \int_0^{t_{\text{life}}} \dot{M}(t) \, dt$$

For a $60 M_\odot$ star, mass loss can reduce the final pre-supernova mass to $\sim 10{-}15 M_\odot$. This dramatically affects the type of supernova (Type Ib/Ic vs Type II) and the mass of the compact remnant.

6. Numerical Simulation: HR Diagram Evolutionary Tracks

We compute approximate evolutionary tracks on the Hertzsprung-Russell diagram for stars of different masses, illustrating the main sequence, red giant branch, horizontal branch, and AGB phases. We also compute the main-sequence lifetime as a function of stellar mass.

HR Diagram Evolutionary Tracks and Main Sequence Lifetimes

Python

script.py342 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ============================================================
# Approximate stellar evolutionary tracks on the HR diagram
# Based on parameterized models of stellar evolution
# ============================================================

def ms_properties(M):
    """Main sequence luminosity, temperature, and radius for mass M (in solar units)."""
    if M < 0.43:
        L = 0.23 * M**2.3
    elif M < 2:
        L = M**4.0
    elif M < 20:
        L = 1.5 * M**3.5
    else:
        L = 3200 * M

if M < 0.7:
        T = 5780 * M**0.5
    elif M < 2:
        T = 5780 * M**0.55
    else:
        T = 5780 * M**0.5

R = M**0.8 if M < 1.5 else M**0.57
    return L, T, R

def generate_track(M_init):
    """Generate an approximate HR diagram track for a star of given initial mass."""
    log_L_track = []
    log_T_track = []

L_ms, T_ms, R_ms = ms_properties(M_init)

# Phase 1: Main Sequence (ZAMS to TAMS)
    n_ms = 50
    for i in range(n_ms):
        frac = i / (n_ms - 1)
        # Stars brighten and cool slightly during MS
        L = L_ms * (1.0 + 0.5 * frac)
        T = T_ms * (1.0 - 0.03 * frac)
        log_L_track.append(np.log10(L))
        log_T_track.append(np.log10(T))

# Phase 2: Subgiant/Red Giant Branch
    if M_init < 0.5:
        # Very low mass: stays on MS forever
        return np.array(log_T_track), np.array(log_L_track)

L_tams = L_ms * 1.5
    T_tams = T_ms * 0.97

if M_init < 8:
        # Low/intermediate mass: red giant branch
        n_rgb = 60
        L_rgb_tip = L_ms * (50 if M_init < 2 else 20) * M_init**0.5
        T_rgb = 3500 + 200 * (M_init - 1)
        T_rgb = max(T_rgb, 3200)
        T_rgb = min(T_rgb, 4500)

for i in range(n_rgb):
            frac = i / (n_rgb - 1)
            # Subgiant: luminosity increases slowly, T drops significantly
            if frac < 0.3:
                f2 = frac / 0.3
                L = L_tams * (1.0 + 2.0 * f2)
                T = T_tams * (1.0 - 0.3 * f2)
            else:
                # RGB: luminosity increases rapidly, T stays low
                f2 = (frac - 0.3) / 0.7
                L = L_tams * 3.0 * (1.0 + (L_rgb_tip / (L_tams * 3) - 1) * f2**1.5)
                T = T_rgb + (T_tams * 0.7 - T_rgb) * (1 - f2)**2

log_L_track.append(np.log10(max(L, 0.01)))
            log_T_track.append(np.log10(max(T, 3000)))

# Phase 3: Horizontal Branch / Red Clump
        L_hb = L_ms * 30 * (M_init / 1.0)**0.3
        T_hb = 4800 if M_init < 2 else 5500
        n_hb = 30

for i in range(n_hb):
            frac = i / (n_hb - 1)
            L = L_hb * (1.0 + 0.3 * frac)
            T = T_hb * (1.0 - 0.1 * frac)
            log_L_track.append(np.log10(max(L, 0.01)))
            log_T_track.append(np.log10(max(T, 3000)))

# Phase 4: AGB
        n_agb = 40
        L_agb_tip = L_rgb_tip * 3
        T_agb = 3200

for i in range(n_agb):
            frac = i / (n_agb - 1)
            L = L_hb * 1.3 * (1.0 + (L_agb_tip / (L_hb * 1.3) - 1) * frac**1.2)
            T = T_hb * 0.9 * (1.0 - 0.35 * frac)
            T = max(T, T_agb)
            log_L_track.append(np.log10(max(L, 0.01)))
            log_T_track.append(np.log10(max(T, 3000)))

# Phase 5: Planetary Nebula to White Dwarf
        if M_init < 8:
            n_pn = 30
            M_wd = 0.08 * M_init + 0.489  # initial-final mass relation
            M_wd = min(M_wd, 1.35)
            L_wd_hot = L_agb_tip * 0.5
            T_wd_hot = 100000
            L_wd_cool = 0.001
            T_wd_cool = 10000

for i in range(n_pn):
                frac = i / (n_pn - 1)
                # Move to high T at roughly constant L, then fade
                if frac < 0.3:
                    f2 = frac / 0.3
                    L = L_agb_tip * (1.0 - 0.5 * f2)
                    T = T_agb * (T_wd_hot / T_agb)**f2
                else:
                    f2 = (frac - 0.3) / 0.7
                    L = L_wd_hot * (L_wd_cool / L_wd_hot)**f2
                    T = T_wd_hot * (T_wd_cool / T_wd_hot)**f2

log_L_track.append(np.log10(max(L, 1e-5)))
                log_T_track.append(np.log10(max(T, 3000)))

else:
        # Massive star evolution
        n_post = 80
        L_rsg = L_ms * 3  # roughly constant L
        T_rsg = 3800

for i in range(n_post):
            frac = i / (n_post - 1)
            if frac < 0.2:
                # Cross to red supergiant
                f2 = frac / 0.2
                L = L_tams * (1.0 + 1.0 * f2)
                T = T_tams * (T_rsg / T_tams)**f2
            elif frac < 0.5:
                # RSG phase
                f2 = (frac - 0.2) / 0.3
                L = L_rsg * (1.0 + 0.5 * f2)
                T = T_rsg * (1.0 + 0.05 * np.sin(f2 * 6))
            elif M_init > 25 and frac < 0.8:
                # WR phase: move back to blue
                f2 = (frac - 0.5) / 0.3
                L = L_rsg * 1.5 * (1.0 - 0.3 * f2)
                T = T_rsg * (50000 / T_rsg)**f2
            else:
                # Pre-SN
                f2 = (frac - max(0.5, 0.8 if M_init > 25 else 0.5)) / (1 - max(0.5, 0.8 if M_init > 25 else 0.5))
                f2 = min(f2, 1.0)
                L = L_rsg * 1.2
                if M_init > 25:
                    T = 50000 * (1.0 - 0.1 * f2)
                else:
                    T = T_rsg * (1.0 + 0.1 * f2)

log_L_track.append(np.log10(max(L, 0.01)))
            log_T_track.append(np.log10(max(T, 3000)))

return np.array(log_T_track), np.array(log_L_track)

# ============================================================
# Create plots
# ============================================================
fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#111111')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('white')
    for spine in ax.spines.values():
        spine.set_color('#333')

# --- Plot 1: Evolutionary tracks for different masses ---
ax1 = axes[0, 0]
masses = [0.8, 1.0, 1.5, 3.0, 5.0, 10.0, 25.0]
track_colors = ['#ff6b35', '#ffa500', '#ffd700', '#88ff44', '#44ddff', '#4488ff', '#aa44ff']

for i, M in enumerate(masses):
    logT, logL = generate_track(M)
    ax1.plot(logT, logL, color=track_colors[i], linewidth=2, label=f'{M} M_sun', alpha=0.9)
    # Mark ZAMS position
    ax1.plot(logT[0], logL[0], 'o', color=track_colors[i], markersize=6)

# Plot ZAMS line
zams_masses = np.logspace(np.log10(0.3), np.log10(60), 100)
zams_logL = []
zams_logT = []
for m in zams_masses:
    L, T, R = ms_properties(m)
    zams_logL.append(np.log10(L))
    zams_logT.append(np.log10(T))
ax1.plot(zams_logT, zams_logL, 'w--', linewidth=1, alpha=0.5, label='ZAMS')

ax1.set_xlabel('log T_eff (K)', fontsize=12)
ax1.set_ylabel('log L/L_sun', fontsize=12)
ax1.set_title('HR Diagram: Evolutionary Tracks', fontsize=14, fontweight='bold')
ax1.legend(fontsize=8, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white', loc='lower left')
ax1.set_xlim(5.2, 3.4)  # reversed: hot on left
ax1.set_ylim(-3, 6.5)
ax1.grid(True, alpha=0.2)

# --- Plot 2: Main sequence lifetime vs mass ---
ax2 = axes[0, 1]
m_arr = np.logspace(np.log10(0.1), np.log10(100), 200)

# MS lifetime: t_MS = eta * f * Mc^2 / L
eta = 0.007
f_fuel = 0.10
t_ms_arr = []
for m in m_arr:
    L, T, R = ms_properties(m)
    t_ms = eta * f_fuel * m / L * 1e10  # in years (normalized to solar)
    t_ms_arr.append(t_ms)
t_ms_arr = np.array(t_ms_arr)

ax2.loglog(m_arr, t_ms_arr, color='#ffa500', linewidth=3)
ax2.axhline(y=1.38e10, color='#555', linestyle=':', linewidth=1)
ax2.text(0.15, 2e10, 'Age of Universe', color='#888', fontsize=9)
ax2.axhline(y=4.6e9, color='#555', linestyle=':', linewidth=1)
ax2.text(0.15, 6e9, 'Age of Sun', color='#888', fontsize=9)

# Mark specific masses
special_m = [0.1, 0.5, 1.0, 3.0, 10.0, 30.0, 100.0]
for m in special_m:
    L, T, R = ms_properties(m)
    t = eta * f_fuel * m / L * 1e10
    ax2.plot(m, t, 'o', color='#ffdd00', markersize=6)
    if m <= 1:
        ax2.text(m*1.2, t*1.3, f'{m}M_sun', color='white', fontsize=8)
    else:
        ax2.text(m*1.2, t*0.7, f'{m}M_sun', color='white', fontsize=8)

ax2.set_xlabel('Mass (M_sun)', fontsize=12)
ax2.set_ylabel('Main Sequence Lifetime (years)', fontsize=12)
ax2.set_title('Main Sequence Lifetime vs Mass', fontsize=14, fontweight='bold')
ax2.set_xlim(0.08, 150)
ax2.set_ylim(1e5, 1e14)
ax2.grid(True, alpha=0.2)

# --- Plot 3: Nuclear burning timescales for massive star ---
ax3 = axes[1, 0]
stages = ['H', 'He', 'C', 'Ne', 'O', 'Si']
# Approximate durations for a 25 Msun star
durations = [7e6, 5e5, 600, 1, 0.5, 0.0025]  # in years
temps = [40, 200, 800, 1500, 2000, 3500]  # in millions of K
colors_stage = ['#ffa500', '#ff8c00', '#ff6347', '#ff4500', '#dc143c', '#8b0000']

bars = ax3.barh(stages, [np.log10(d) for d in durations], color=colors_stage,
                edgecolor='white', linewidth=0.5, alpha=0.85)
ax3.set_xlabel('log10(Duration in years)', fontsize=12)
ax3.set_title('Nuclear Burning Timescales (25 M_sun)', fontsize=14, fontweight='bold')
ax3.set_xlim(-3, 8)
ax3.grid(True, alpha=0.2, axis='x')

for bar, d, T in zip(bars, durations, temps):
    if d >= 1:
        label = f'{d:.0f} yr, T ~ {T} MK'
    elif d >= 0.01:
        label = f'{d*365:.0f} days, T ~ {T} MK'
    else:
        label = f'{d*365*24:.0f} hr, T ~ {T} MK'
    ax3.text(bar.get_width() + 0.1, bar.get_y() + bar.get_height()/2,
             label, va='center', color='white', fontsize=9)

# --- Plot 4: Initial-Final mass relation ---
ax4 = axes[1, 1]
m_init = np.linspace(0.8, 8.0, 100)
# Empirical initial-final mass relation
m_final = 0.08 * m_init + 0.489
m_final = np.clip(m_final, 0.5, 1.38)

# Massive stars
m_init_massive = np.linspace(8, 60, 100)
m_remnant = np.where(m_init_massive < 25, 1.4, 1.4 + 0.1 * (m_init_massive - 25))  # NS mass
m_bh = np.where(m_init_massive > 25, m_init_massive * 0.15, 0)  # BH mass estimate

ax4.plot(m_init, m_final, color='#ffa500', linewidth=2.5, label='White Dwarf')
ax4.plot(m_init_massive[:50], np.ones(50) * 1.4, color='#44ddff', linewidth=2.5, label='Neutron Star')
ax4.plot(m_init_massive[50:], m_bh[50:], color='#aa44ff', linewidth=2.5, label='Black Hole')

ax4.axhline(y=1.44, color='#ff4444', linestyle='--', linewidth=1, alpha=0.7)
ax4.text(1.0, 1.48, 'Chandrasekhar limit', color='#ff4444', fontsize=9)
ax4.axvline(x=8.0, color='#555', linestyle=':', linewidth=1)
ax4.text(8.3, 0.3, 'M ~ 8 M_sun', color='#888', fontsize=9)

ax4.set_xlabel('Initial Mass (M_sun)', fontsize=12)
ax4.set_ylabel('Remnant Mass (M_sun)', fontsize=12)
ax4.set_title('Initial-Final Mass Relation', fontsize=14, fontweight='bold')
ax4.legend(fontsize=10, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white')
ax4.set_xlim(0.5, 60)
ax4.set_ylim(0, 10)
ax4.grid(True, alpha=0.2)

# Print key results
print("=" * 65)
print("STELLAR EVOLUTION: KEY RESULTS")
print("=" * 65)

print("\nMain Sequence Lifetimes:")
print(f"{'Mass (Msun)':>15} {'Lifetime':>20} {'Spectral Type':>15}")
print("-" * 55)
mass_spec = [(0.1, 'M8'), (0.5, 'M0'), (1.0, 'G2'), (1.5, 'F5'),
             (3.0, 'A0'), (10.0, 'B2'), (30.0, 'O7'), (60.0, 'O4')]
for m, sp in mass_spec:
    L, T, R = ms_properties(m)
    t = eta * f_fuel * m / L * 1e10
    if t > 1e9:
        t_str = f'{t/1e9:.1f} Gyr'
    elif t > 1e6:
        t_str = f'{t/1e6:.1f} Myr'
    else:
        t_str = f'{t/1e3:.0f} kyr'
    print(f'{m:>15.1f} {t_str:>20} {sp:>15}')

print("\nNuclear burning stages for 25 Msun star:")
for stage, dur, T in zip(stages, durations, temps):
    if dur >= 1:
        print(f"  {stage:>3} burning: {dur:>12.1f} years  (T ~ {T} MK)")
    else:
        print(f"  {stage:>3} burning: {dur*365*24:>12.1f} hours  (T ~ {T} MK)")

print("\nChandrasekhar mass: 1.44 Msun (for mu_e = 2)")
print("\nFinal fates:")
print("  M < 8 Msun:  White dwarf (via AGB mass loss)")
print("  8-25 Msun:   Neutron star (via core-collapse SN)")
print("  M > 25 Msun: Black hole (direct collapse or fallback)")

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
print("\nPlot saved to output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Chapter Summary

The main-sequence lifetime scales as $t_{\text{MS}} \approx 10^{10}(M/M_\odot)^{-2.5}$ years, ranging from trillions of years for red dwarfs to millions of years for O-type stars. Post-main-sequence evolution is governed by the mirror principle: as burning shells cause the core to contract, the envelope expands, creating red giants.

The Chandrasekhar mass $M_{\text{Ch}} = 1.44 M_\odot$ sets the maximum mass for electron-degenerate white dwarfs. Cores exceeding this limit undergo catastrophic collapse, driven by iron photodisintegration and electron capture, producing neutron stars or black holes.

Mass loss through radiation-driven winds, AGB superwinds, and WR-star outflows critically determines the final remnant mass and is responsible for enriching the interstellar medium with heavy elements.

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