Part V: Structure Formation | Chapter 2

Stellar Classification & the Hertzsprung-Russell Diagram

Understanding stellar populations through spectral classification, the HR diagram, stellar structure equations, and evolutionary pathways from the main sequence to compact remnants

1. Spectral Classification (OBAFGKM)

The modern classification of stars by their spectra dates to the work of Annie Jump Cannon at Harvard in the early 20th century. The Morgan-Keenan (MK) system arranges stars into spectral classes ordered by decreasing surface temperature: O, B, A, F, G, K, M. The famous mnemonic “Oh Be A Fine Girl/Guy, Kiss Me” encodes this sequence. Each class is further subdivided with a digit 0–9 (e.g., G2 for the Sun).

Spectral Classes and Their Properties

Class	T_eff (K)	Color	Key Spectral Features	Example
O	30,000–50,000+	Blue	He II, N III, C III ionized lines	10 Lacertae
B	10,000–30,000	Blue-white	He I, H Balmer (strong)	Rigel, Spica
A	7,500–10,000	White	H Balmer (maximum), some metals	Sirius, Vega
F	6,000–7,500	Yellow-white	Ca II (H & K), weakening H	Canopus, Procyon
G	5,200–6,000	Yellow	Ca II (strong), Fe I, CH G-band	Sun (G2V)
K	3,700–5,200	Orange	Ca I, Fe I, molecular bands begin	Arcturus, Aldebaran
M	2,400–3,700	Red	TiO, VO molecular bands dominate	Betelgeuse, Proxima Cen

1.1 Wien's Displacement Law

The peak wavelength of a star's blackbody emission is directly related to its surface temperature through Wien's displacement law:

$$\lambda_{\max} = \frac{b}{T}, \qquad b = 2.898 \times 10^{-3} \;\text{m}\cdot\text{K}$$

For the Sun ($T_{\text{eff}} = 5778$ K), this gives $\lambda_{\max} \approx 501$ nm (green-yellow), while an O-type star at 40,000 K peaks at $\lambda_{\max} \approx 72$ nm (far ultraviolet).

1.2 Luminosity Classes

The MK system adds a luminosity class denoting the evolutionary state and surface gravity of the star. Luminosity classes are determined from the width of spectral lines — lower surface gravity in giants and supergiants produces narrower, sharper lines due to reduced pressure broadening:

Ia, Ib — Luminous and less-luminous supergiants ($L \sim 10^4 - 10^6 \, L_\odot$)
II — Bright giants
III — Normal giants ($L \sim 10^1 - 10^3 \, L_\odot$)
IV — Subgiants (transitioning off the main sequence)
V — Main sequence (dwarfs) — the Sun is classified G2V

2. The Hertzsprung-Russell Diagram

2.1 Historical Development

In 1911, the Danish astronomer Ejnar Hertzsprung plotted the absolute magnitudes of stars in the Pleiades and Hyades clusters against their spectral types, discovering that most stars fell along a diagonal band. Independently, the American astronomer Henry Norris Russell presented a similar diagram in 1913 using nearby stars with known parallaxes. The resulting Hertzsprung-Russell (HR) diagram became one of the most important tools in all of astrophysics — a Rosetta Stone connecting observable properties to the underlying physics of stellar structure and evolution.

2.2 Axes and Layout

The HR diagram plots luminosity (in solar units, $L/L_\odot$) on the vertical axis against effective temperature $T_{\text{eff}}$ on the horizontal axis. By convention, temperature increases to the left, so hot blue stars appear on the left and cool red stars on the right. The main features of the diagram are:

Main Sequence — A diagonal band running from hot, luminous O-stars (upper left) to cool, faint M-dwarfs (lower right). About 90% of all stars reside here, burning hydrogen in their cores.
Red Giant Branch (RGB) — Upper right region populated by evolved stars with exhausted hydrogen cores but hydrogen-burning shells. Radii of $\sim 10 - 100 \, R_\odot$.
Horizontal Branch (HB) — Stars burning helium in their cores after the helium flash, approximately constant luminosity at $L \sim 50 \, L_\odot$.
Asymptotic Giant Branch (AGB) — Double-shell-burning stars ascending the giant branch a second time, luminosities up to $\sim 10^4 \, L_\odot$.
White Dwarf Region — Lower left, hot but extremely faint due to tiny radii ($R \sim R_\oplus$). These are the exposed cores of low/intermediate mass stars.

2.3 The Mass-Luminosity Relation

For main-sequence stars, there is a tight relationship between mass and luminosity that follows from the physics of stellar interiors:

$$L \propto M^{\alpha}, \qquad \alpha \approx \begin{cases} 2.3 & M < 0.43 \, M_\odot \\ 4.0 & 0.43 \leq M < 2 \, M_\odot \\ 3.5 & 2 \leq M < 55 \, M_\odot \\ 1.0 & M > 55 \, M_\odot \end{cases}$$

The steep dependence means a star twice as massive as the Sun is roughly $2^4 = 16$ times more luminous, drastically shortening its lifetime.

2.4 The Stefan-Boltzmann Law

A star's luminosity, radius, and effective temperature are connected by the Stefan-Boltzmann law:

$$L = 4\pi R^2 \sigma T_{\text{eff}}^4$$

where $\sigma = 5.670 \times 10^{-8}$ W m$^{-2}$ K$^{-4}$ is the Stefan-Boltzmann constant. Lines of constant radius on the HR diagram are diagonal lines from upper-left to lower-right.

2.5 Magnitudes and Color Index

Astronomers often work with magnitudes rather than luminosities. The distance modulus relates apparent magnitude $m$ and absolute magnitude $M$ to the distance $d$:

$$m - M = 5 \log_{10}\!\left(\frac{d}{10 \;\text{pc}}\right)$$

The color index $B - V$ measures the difference in apparent magnitude between the blue (B) and visual (V) filters. Hot O/B stars have $B - V \approx -0.3$, the Sun has $B - V = 0.65$, and cool M-stars have $B - V \approx 1.5 - 2.0$.

3. Stellar Structure Equations

The internal structure of a star in hydrostatic equilibrium is governed by four coupled differential equations, supplemented by an equation of state and opacity law. These equations determine how pressure, mass, luminosity, and temperature vary as functions of radius.

3.1 Hydrostatic Equilibrium

At every shell within the star, the inward pull of gravity must be balanced by the outward pressure gradient:

$$\frac{dP}{dr} = -\frac{G \rho(r) \, m(r)}{r^2}$$

This is the fundamental equation of stellar structure. Violation of this balance leads to dynamical collapse (free-fall timescale $\tau_{\rm ff} \sim (G\bar{\rho})^{-1/2}$) or explosion.

3.2 Mass Continuity

$$\frac{dm}{dr} = 4\pi r^2 \rho(r)$$

The mass enclosed within radius $r$ increases with each spherical shell of density $\rho$.

3.3 Radiative Energy Transport

In radiative zones, photons carry energy outward through a temperature gradient determined by the opacity $\kappa$:

$$\frac{dT}{dr} = -\frac{3 \kappa \rho \, L(r)}{16\pi a c \, r^2 T^3}$$

where $a = 4\sigma/c$ is the radiation constant. High opacity or luminosity steepens the temperature gradient. If the gradient becomes too steep, convection sets in (Schwarzschild criterion).

3.4 Energy Generation

$$\frac{dL}{dr} = 4\pi r^2 \rho(r) \, \varepsilon(r)$$

where $\varepsilon$ is the energy generation rate per unit mass. Nuclear reactions are the dominant energy source on the main sequence.

3.5 Equation of State and Nuclear Reactions

For main-sequence stars, the equation of state is well-approximated by an ideal gas:

$$P = \frac{\rho k_B T}{\mu m_H}$$

where $\mu \approx 0.62$ is the mean molecular weight for solar composition.

The two dominant hydrogen-burning pathways have dramatically different temperature sensitivities:

pp-chain (dominant for $M \lesssim 1.3 \, M_\odot$):$$\varepsilon_{\rm pp} \propto \rho \, T^4$$Moderate temperature sensitivity; operates in the Sun's core at $T_c \approx 1.57 \times 10^7$ K.
CNO cycle (dominant for $M \gtrsim 1.3 \, M_\odot$):$$\varepsilon_{\rm CNO} \propto \rho \, T^{16}$$Extremely steep temperature dependence, making the CNO cycle the dominant energy source in massive stars. The high sensitivity drives convective cores.

4. Stellar Evolution

4.1 Main Sequence Lifetime

A star's time on the main sequence is set by its fuel supply (proportional to mass) divided by its fuel consumption rate (luminosity):

$$\tau_{\rm MS} \approx 10^{10} \; \frac{M/M_\odot}{L/L_\odot} \;\text{years} \approx 10^{10} \left(\frac{M}{M_\odot}\right)^{1-\alpha} \;\text{years}$$

The Sun has $\tau_{\rm MS} \approx 10$ Gyr. A 10 $M_\odot$ star with $L \sim 10^{3.5} L_\odot$ lasts only $\sim 20$ Myr, while a 0.1 $M_\odot$ red dwarf can burn for trillions of years.

4.2 Post-Main-Sequence Evolution by Mass

Low Mass ($M < 0.5 \, M_\odot$): Red Dwarfs

Fully convective, so they mix hydrogen throughout and burn nearly all of it. Main sequence lifetimes exceed the current age of the universe. Eventually contract to become helium white dwarfs, though none have yet formed in nature.

Intermediate Mass ($0.5 - 8 \, M_\odot$)

This is the evolutionary path followed by the Sun and similar stars:

Main Sequence → core hydrogen exhaustion
Red Giant Branch (RGB) → hydrogen shell burning, degenerate He core grows
Helium Flash (for $M \lesssim 2 M_\odot$) → explosive ignition of degenerate He core
Horizontal Branch → stable core He burning + H shell burning
Asymptotic Giant Branch (AGB) → double shell burning, thermal pulses
Planetary Nebula → envelope ejection, exposing the hot core
White Dwarf → C/O degenerate remnant, slowly cooling

High Mass ($M > 8 \, M_\odot$): Massive Stars

Massive stars burn through successive nuclear fuels (H, He, C, Ne, O, Si) in an onion-shell structure. The final iron core cannot release energy through fusion. When the core exceeds the Chandrasekhar mass, electron degeneracy pressure fails:

Core-collapse supernova — the iron core collapses in milliseconds, rebounding into a powerful shock wave
Neutron star ($M_{\rm remnant} \lesssim 2-3 \, M_\odot$) — supported by neutron degeneracy pressure
Black hole ($M_{\rm remnant} \gtrsim 2-3 \, M_\odot$) — complete gravitational collapse

4.3 Critical Mass Limits

Chandrasekhar Limit

$$M_{\rm Ch} = \frac{5.83}{\mu_e^2} \, M_\odot \approx 1.44 \, M_\odot$$

Maximum mass of a white dwarf supported by electron degeneracy pressure ($\mu_e = 2$for C/O composition).

Tolman-Oppenheimer-Volkoff Limit

$$M_{\rm TOV} \approx 2.0 - 3.0 \, M_\odot$$

Maximum mass of a neutron star. The exact value depends on the nuclear equation of state, which remains an active area of research. Observations of massive pulsars ($\sim 2.0 \, M_\odot$) constrain the lower bound.

5. Python Simulation: Interactive HR Diagram

The following Python script generates a synthetic Hertzsprung-Russell diagram with approximately 1,000 stars distributed across the main sequence, red giant branch, white dwarf region, and supergiant population. Notable stars are marked and labeled, and the spectral class boundaries are indicated along the temperature axis.

Synthetic HR Diagram Generator

Python

Generates a detailed HR diagram with ~1000 stars across multiple evolutionary populations, proper color mapping by temperature, and labeled notable stars.

script.py163 lines

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.colors import LogNorm

np.random.seed(42)

fig, ax = plt.subplots(1, 1, figsize=(12, 9))
fig.patch.set_facecolor('#0f0a1a')
ax.set_facecolor('#0f0a1a')

# ============================================
# Helper: temperature -> RGB color
# ============================================
def temp_to_rgb(T):
    """Approximate blackbody color for plotting."""
    if T > 30000:
        return (0.6, 0.7, 1.0)
    elif T > 10000:
        frac = (T - 10000) / 20000.0
        return (0.7 + 0.3*(1-frac), 0.8 + 0.2*(1-frac), 1.0)
    elif T > 7500:
        frac = (T - 7500) / 2500.0
        return (1.0, 1.0, 0.85 + 0.15*frac)
    elif T > 6000:
        frac = (T - 6000) / 1500.0
        return (1.0, 1.0, 0.6 + 0.25*frac)
    elif T > 5000:
        frac = (T - 5000) / 1000.0
        return (1.0, 0.85 + 0.15*frac, 0.3 + 0.3*frac)
    elif T > 3500:
        frac = (T - 3500) / 1500.0
        return (1.0, 0.6 + 0.25*frac, 0.1 + 0.2*frac)
    else:
        return (1.0, 0.4, 0.1)

# ============================================
# 1. Main Sequence Stars (~700 stars)
# ============================================
N_ms = 700
log_mass_ms = np.random.uniform(np.log10(0.08), np.log10(80), N_ms)
mass_ms = 10**log_mass_ms

# Mass-luminosity: L/Lsun ~ M^3.5 (with scatter)
alpha = np.where(mass_ms < 0.43, 2.3,
        np.where(mass_ms < 2.0, 4.0,
        np.where(mass_ms < 55, 3.5, 1.0)))
log_L_ms = alpha * np.log10(mass_ms) + np.random.normal(0, 0.15, N_ms)

# Mass-temperature: T ~ 5778 * M^0.57
log_T_ms = np.log10(5778) + 0.57 * np.log10(mass_ms) + np.random.normal(0, 0.03, N_ms)

T_ms = 10**log_T_ms
L_ms = 10**log_L_ms

# ============================================
# 2. Red Giant Branch (~150 stars)
# ============================================
N_rgb = 150
T_rgb = 10**(np.random.uniform(np.log10(3200), np.log10(5200), N_rgb))
L_rgb = 10**(np.random.uniform(1.0, 3.5, N_rgb))
# Add correlation: cooler giants are more luminous on AGB
L_rgb *= (5200.0 / T_rgb)**0.5

# ============================================
# 3. White Dwarfs (~100 stars)
# ============================================
N_wd = 100
T_wd = 10**(np.random.uniform(np.log10(5000), np.log10(40000), N_wd))
# L ~ 10^{-2} to 10^{-4.5} Lsun
L_wd = 10**(np.random.uniform(-4.5, -1.5, N_wd))

# ============================================
# 4. Supergiants (~50 stars)
# ============================================
N_sg = 50
T_sg = 10**(np.random.uniform(np.log10(3500), np.log10(30000), N_sg))
L_sg = 10**(np.random.uniform(3.5, 5.8, N_sg))

# ============================================
# Combine all populations
# ============================================
all_T = np.concatenate([T_ms, T_rgb, T_wd, T_sg])
all_L = np.concatenate([L_ms, L_rgb, L_wd, L_sg])

# Plot each star with its color
for T, L in zip(all_T, all_L):
    color = temp_to_rgb(T)
    ax.scatter(T, L, c=[color], s=8, alpha=0.7, edgecolors='none')

# ============================================
# 5. Mark notable stars
# ============================================
notable = {
    'Sun':              (5778,   1.0,       'gold',   12),
    'Sirius A':         (9940,   25.4,      'cyan',   12),
    'Betelgeuse':       (3500,   126000,    'red',    14),
    'Rigel':            (12100,  120000,    'deepskyblue', 14),
    'Proxima Centauri': (3042,   0.0017,    'orangered',   12),
    'Vega':             (9602,   40.12,     'lightcyan',   11),
    'Sirius B':         (25200,  0.056,     'white',  10),
}

for name, (T, L, c, sz) in notable.items():
    ax.scatter(T, L, c=c, s=sz**2, edgecolors='white',
               linewidths=0.5, zorder=10)
    offset = (15, 10) if name != 'Sirius B' else (15, -15)
    ax.annotate(name, (T, L), textcoords='offset points',
                xytext=offset, fontsize=8, color=c,
                fontweight='bold',
                arrowprops=dict(arrowstyle='->', color=c, lw=0.8))

# ============================================
# 6. Spectral class labels along top
# ============================================
spec_classes = [
    ('O', 40000), ('B', 20000), ('A', 8750),
    ('F', 6750),  ('G', 5600),  ('K', 4450), ('M', 3200)
]
for label, T in spec_classes:
    ax.annotate(label, xy=(T, ax.get_ylim()[1] if ax.get_ylim()[1] > 0 else 1e6),
                xytext=(T, 2e6), fontsize=14, fontweight='bold',
                color='#c4b5fd', ha='center', va='bottom')

# ============================================
# 7. Region labels
# ============================================
ax.text(4200, 500, 'Red Giants', fontsize=11, color='#fca5a5',
        fontstyle='italic', alpha=0.8)
ax.text(15000, 1e5, 'Supergiants', fontsize=11, color='#c4b5fd',
        fontstyle='italic', alpha=0.8)
ax.text(5500, 3e-2, 'Main Sequence', fontsize=11, color='#fde68a',
        fontstyle='italic', alpha=0.8, rotation=-35)
ax.text(15000, 5e-4, 'White Dwarfs', fontsize=11, color='#e0e0e0',
        fontstyle='italic', alpha=0.8)

# ============================================
# Formatting
# ============================================
ax.set_xscale('log')
ax.set_yscale('log')
ax.invert_xaxis()
ax.set_xlim(50000, 2500)
ax.set_ylim(1e-5, 5e6)
ax.set_xlabel('Effective Temperature (K)', fontsize=14, color='white')
ax.set_ylabel(r'Luminosity ($L/L_{\odot}$)', fontsize=14, color='white')
ax.set_title('Hertzsprung-Russell Diagram', fontsize=18,
             color='#e9d5ff', fontweight='bold', pad=20)
ax.tick_params(colors='white', labelsize=11)
for spine in ax.spines.values():
    spine.set_color('#7c3aed')
    spine.set_linewidth(1.5)
ax.grid(True, alpha=0.15, color='purple')

plt.tight_layout()
plt.savefig('hr_diagram.png', dpi=150, bbox_inches='tight',
            facecolor='#0f0a1a')
plt.show()
print("HR diagram saved to hr_diagram.png")
print(f"Total stars plotted: {len(all_T)}")
print(f"  Main sequence: {N_ms}")
print(f"  Red giants:    {N_rgb}")
print(f"  White dwarfs:  {N_wd}")
print(f"  Supergiants:   {N_sg}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Expected Output

The script produces a publication-quality HR diagram on a dark background with:

Main sequence band: 700 stars with realistic scatter from mass-luminosity and mass-temperature relations
Red giant branch: 150 stars in the upper-right, with cooler giants being more luminous
White dwarfs: 100 stars in the lower-left, hot but faint
Supergiants: 50 stars across the top of the diagram
Notable stars: Sun, Sirius A, Betelgeuse, Rigel, Proxima Centauri, Vega, and Sirius B annotated with arrows

6. Fortran Simulation: Stellar Structure

This Fortran program solves the Lane-Emden equation for an $n = 3$ polytrope (appropriate for radiation-pressure-supported stellar interiors) and uses scaling relations to compute physical properties — central temperature, central pressure, effective temperature, and main sequence lifetime — for stars ranging from 0.1 to 80 solar masses.

Polytropic Stellar Structure Solver

Fortran

Solves the Lane-Emden equation for n=3 polytrope and computes central temperature, pressure, luminosity, and main sequence lifetime for stars of different masses.

stellar_structure.f90152 lines

program stellar_structure
    implicit none

! =============================================
    ! Polytropic Stellar Structure Solver
    ! Solves Lane-Emden equation for n=3 polytrope
    ! and computes stellar properties for
    ! different masses using scaling relations.
    ! =============================================

integer, parameter :: dp = selected_real_kind(15, 307)
    integer, parameter :: N_steps = 100000
    real(dp), parameter :: pi = 3.141592653589793_dp

! Physical constants (CGS)
    real(dp), parameter :: G     = 6.67430e-8_dp    ! grav constant
    real(dp), parameter :: c_light = 2.99792e10_dp  ! speed of light
    real(dp), parameter :: k_B   = 1.38065e-16_dp   ! Boltzmann
    real(dp), parameter :: m_H   = 1.67353e-24_dp   ! proton mass
    real(dp), parameter :: sigma_SB = 5.6704e-5_dp  ! Stefan-Boltzmann
    real(dp), parameter :: Msun  = 1.989e33_dp       ! solar mass
    real(dp), parameter :: Lsun  = 3.828e33_dp       ! solar luminosity
    real(dp), parameter :: Rsun  = 6.957e10_dp       ! solar radius

! Lane-Emden variables
    real(dp) :: xi, theta, dtheta, d2theta
    real(dp) :: dxi, xi1, dtheta1
    integer  :: i

! Stellar models
    integer, parameter :: N_stars = 12
    real(dp) :: masses(N_stars), luminosities(N_stars)
    real(dp) :: radii(N_stars), T_eff(N_stars)
    real(dp) :: T_central(N_stars), P_central(N_stars)
    real(dp) :: tau_ms(N_stars)
    real(dp) :: M, L, R, Tc, Pc, alpha_ML
    integer  :: j

! =============================================
    ! Step 1: Solve Lane-Emden for n=3
    ! =============================================
    dxi = 1.0e-4_dp
    xi = dxi
    theta = 1.0_dp - xi**2 / 6.0_dp
    dtheta = -xi / 3.0_dp

do i = 1, N_steps
        if (theta <= 0.0_dp) exit
        d2theta = -theta**3 - 2.0_dp * dtheta / xi
        theta  = theta  + dtheta  * dxi
        dtheta = dtheta + d2theta * dxi
        xi = xi + dxi
    end do

xi1 = xi
    dtheta1 = dtheta

write(*,'(A)') '=============================================='
    write(*,'(A)') ' Lane-Emden Solution for n=3 Polytrope'
    write(*,'(A)') '=============================================='
    write(*,'(A,F10.6)') ' xi_1 (surface)       = ', xi1
    write(*,'(A,F10.6)') ' -xi1^2 * dtheta(xi1) = ', -xi1**2 * dtheta1
    write(*,'(A)') ''

! =============================================
    ! Step 2: Stellar models for different masses
    ! =============================================
    masses = (/ 0.1_dp, 0.3_dp, 0.5_dp, 0.8_dp, 1.0_dp, &
                1.5_dp, 2.0_dp, 5.0_dp, 10.0_dp, 20.0_dp, &
                40.0_dp, 80.0_dp /)

write(*,'(A)') '=============================================='
    write(*,'(A)') ' Main Sequence Stellar Properties'
    write(*,'(A)') '=============================================='
    write(*,'(A)') ''
    write(*,'(A5, A12, A12, A12, A14, A14, A14)') &
        'M/Ms', 'L/Ls', 'R/Rs', 'Teff(K)', &
        'Tc(MK)', 'Pc(dyn)', 'tau_MS(Gyr)'

write(*,'(A)') '----------------------------------------------' // &
        '----------------------------------------------'

do j = 1, N_stars
        M = masses(j) * Msun

! Mass-luminosity relation (piecewise)
        if (masses(j) < 0.43_dp) then
            alpha_ML = 2.3_dp
        else if (masses(j) < 2.0_dp) then
            alpha_ML = 4.0_dp
        else if (masses(j) < 55.0_dp) then
            alpha_ML = 3.5_dp
        else
            alpha_ML = 1.0_dp
        end if
        L = masses(j)**alpha_ML * Lsun
        luminosities(j) = L / Lsun

! Mass-radius: R ~ M^0.8 for M > 1, R ~ M^0.57 for M < 1
        if (masses(j) < 1.0_dp) then
            R = masses(j)**0.8_dp * Rsun
        else
            R = masses(j)**0.57_dp * Rsun
        end if
        radii(j) = R / Rsun

! Effective temperature from Stefan-Boltzmann
        T_eff(j) = (L / (4.0_dp * pi * R**2 * sigma_SB))**0.25_dp

! Central temperature (polytropic scaling)
        ! Tc ~ (G * mu * mH / kB) * (M / R) * polytropic_factor
        Tc = 0.4_dp * G * 0.62_dp * m_H * M / (k_B * R)
        T_central(j) = Tc

! Central pressure
        Pc = 0.77_dp * G * M**2 / (R**4 * 4.0_dp * pi)
        P_central(j) = Pc

! Main sequence lifetime: tau ~ (M/L) * tau_sun
        ! tau_sun ~ 10 Gyr
        tau_ms(j) = 10.0_dp * masses(j) / luminosities(j)

write(*,'(F5.1, ES12.3, F12.3, F12.0, F14.2, ES14.3, F14.4)') &
            masses(j), luminosities(j), radii(j), T_eff(j), &
            Tc / 1.0e6_dp, Pc, tau_ms(j)
    end do

write(*,'(A)') ''
    write(*,'(A)') '=============================================='
    write(*,'(A)') ' Mass-Luminosity Relation Verification'
    write(*,'(A)') '=============================================='
    write(*,'(A)') ' Fitting L = M^alpha:'
    write(*,'(A)') '   M < 0.43 Msun:  alpha = 2.3'
    write(*,'(A)') '   0.43-2 Msun:    alpha = 4.0'
    write(*,'(A)') '   2-55 Msun:      alpha = 3.5'
    write(*,'(A)') '   >55 Msun:       alpha = 1.0'
    write(*,'(A)') ''

! =============================================
    ! Step 3: Chandrasekhar and TOV limits
    ! =============================================
    write(*,'(A)') '=============================================='
    write(*,'(A)') ' Stellar Remnant Mass Limits'
    write(*,'(A)') '=============================================='
    write(*,'(A,F6.3,A)') ' Chandrasekhar limit: ', 1.44_dp, ' Msun'
    write(*,'(A,A)')       ' TOV limit:           ', '~2.0-3.0 Msun'
    write(*,'(A)') ''
    write(*,'(A)') ' Stars < 8 Msun  -> White Dwarf (< Mch)'
    write(*,'(A)') ' Stars 8-25 Msun -> Neutron Star'
    write(*,'(A)') ' Stars > 25 Msun -> Black Hole'

end program stellar_structure

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Expected Output

The program outputs:

Lane-Emden solution: $\xi_1 \approx 6.897$ for $n = 3$
Stellar properties table: mass, luminosity, radius, effective temperature, central temperature, central pressure, and main sequence lifetime for 12 representative masses
Mass-luminosity verification: confirming the piecewise power-law exponents
Remnant mass limits: Chandrasekhar (1.44 $M_\odot$) and TOV (~2–3 $M_\odot$)

7. Video Resources

The Hertzsprung-Russell Diagram Explained

Professor Dave Explains provides a clear overview of how the HR diagram is constructed and what the different regions tell us about stellar properties.

Stellar Evolution: Life Cycle of a Star

A comprehensive overview of stellar evolution from protostellar collapse through the main sequence to white dwarfs, neutron stars, and black holes.

Spectral Classification of Stars

An educational walkthrough of the OBAFGKM spectral classification system, including how absorption lines arise from atomic physics and what they reveal about stellar atmospheres.

MIT 8.901 — Stellar Structure and Evolution

A lecture from MIT's graduate astrophysics course covering the equations of stellar structure, polytropic models, and the physics underlying the mass-luminosity relation.

8. Summary

Key Equations of Stellar Astrophysics

Wien's Displacement Law

$$\lambda_{\max} = \frac{2.898 \times 10^{-3} \;\text{m}\cdot\text{K}}{T}$$

Stefan-Boltzmann Law

$$L = 4\pi R^2 \sigma T_{\text{eff}}^4$$

Hydrostatic Equilibrium

$$\frac{dP}{dr} = -\frac{G \rho \, m(r)}{r^2}$$

Mass-Luminosity Relation

$$L \propto M^{3.5-4} \quad \text{(main sequence)}$$

Main Sequence Lifetime

$$\tau_{\rm MS} \approx 10^{10} \;\frac{M/M_\odot}{L/L_\odot} \;\text{years}$$

Chandrasekhar Limit

$$M_{\rm Ch} \approx 1.44 \, M_\odot$$

The Physical Picture

1. Stars are classified by the OBAFGKM spectral sequence, ordered by decreasing temperature from >30,000 K (O-type) to <3,700 K (M-type), with luminosity classes I–V distinguishing supergiants from dwarfs.

2. The HR diagram reveals that ~90% of stars lie on the main sequence, a tight band in luminosity-temperature space governed by the mass-luminosity relation $L \propto M^{3.5\text{-}4}$.

3. Stellar interiors are described by four structure equations (hydrostatic equilibrium, mass continuity, energy transport, energy generation) coupled with an equation of state.

4. The pp-chain ($\varepsilon \propto T^4$) dominates in solar-type stars, while the CNO cycle ($\varepsilon \propto T^{16}$) drives convective cores in massive stars.

5. Post-main-sequence evolution depends critically on mass: low-mass stars become white dwarfs (below the Chandrasekhar limit of 1.44 $M_\odot$), while massive stars undergo core collapse to form neutron stars or black holes.

6. The steep mass-luminosity relation means massive stars live fast and die young ($\sim$Myr), while red dwarfs can burn for trillions of years — far exceeding the current age of the universe.

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