E5-3. Hertzsprung-Russell (HR) Diagram & Stellar Spectra HL

1. The Hertzsprung-Russell Diagram

The Hertzsprung-Russell (HR) diagram is a monumental scatter plot of stars showing the relationship between their absolute Luminosity (relative to the Sun, $L/L_\odot$) and their surface Temperature ($T$ in Kelvin).

  • Axes Anomalies: The x-axis (Temperature) is plotted backwards, from hot (left) to cool (right). Both axes are usually logarithmic.
  • Main Sequence: A prominent diagonal band running from top-left (hot, bright, massive stars) to bottom-right (cool, dim, low-mass stars). Stars here are stably fusing Hydrogen into Helium.
  • Red Giants & Supergiants: Located in the top-right. They are cool (red) but incredibly luminous because of their massive surface area.
  • White Dwarfs: Located in the bottom-left. They are very hot (white) but incredibly dim because they are exceptionally small (Earth-sized).
Hertzsprung-Russell Diagram Luminosity (L / L_sun) 10⁶ 10⁴ 10² 1 10⁻² 10⁻⁴ Surface Temperature (K) 40,000 10,000 6,000 4,000 2,500 Main Sequence Sun Supergiants Betelgeuse Red Giants Aldebaran White Dwarfs Sirius B

2. Stellar Spectra & Wien's Displacement Law

  • Stars approximate Black Body Radiators. They emit a continuous spectrum of electromagnetic radiation across all wavelengths, peaking at a specific wavelength ($\lambda_{max}$) which determines their apparent color.
  • Wien's Displacement Law: The peak wavelength of emission is inversely proportional to the absolute surface temperature of the star.
    $$\lambda_{max} T = 2.9 \times 10^{-3}\text{ m K}$$
  • As the continuous spectrum generated by the hot, dense core passes through the cooler, low-pressure outer atmosphere of the star, specific wavelengths are absorbed by gaseous elements, creating dark Absorption Spectra lines. These spectral fingerprints allow astronomers to identify the chemical composition of distant stars.

3. Advanced HL Examples

Example 1: Utilizing Wien's Displacement Law

Problem: The star Rigel appears bright blue-white in the sky, while Antares appears distinctly reddish-orange. A telescope spectrometer determines that the peak wavelength of radiation emitted by Rigel is $\lambda_R = 240\text{ nm}$, and for Antares it is $\lambda_A = 850\text{ nm}$.
Calculate the surface temperatures of both stars, and confirm which star belongs on the far left side of the HR Diagram.

Solution:

Step 1: Apply Wien's Law for Rigel.
$$T = \frac{2.9 \times 10^{-3}}{\lambda_{max}}$$ $$T_R = \frac{2.9 \times 10^{-3}\text{ m K}}{240 \times 10^{-9}\text{ m}} = \mathbf{12,083\text{ K}}$$

Step 2: Apply Wien's Law for Antares.
$$T_A = \frac{2.9 \times 10^{-3}\text{ m K}}{850 \times 10^{-9}\text{ m}} = \mathbf{3,412\text{ K}}$$

Conclusion: Rigel ($\approx 12,000\text{ K}$) is significantly hotter than Antares ($\approx 3,400\text{ K}$). Therefore, Rigel belongs on the far left of the HR Diagram (high temperature, short peak wavelength), and Antares belongs on the right side.

Example 2: Determining Radii directly from HR Diagram Coordinates

Problem: By combining the Stefan-Boltzmann Law ($L = 4\pi R^2 \sigma T^4$) for a star and the Sun, we can derive a scaling ratio: $\frac{L}{L_\odot} = \left(\frac{R}{R_\odot}\right)^2 \left(\frac{T}{T_\odot}\right)^4$.
A red supergiant is plotted on the HR diagram with a luminosity of $L = 10^5 L_\odot$ and a surface temperature of $T = 3000\text{ K}$. Assuming the Sun has a surface temperature of $T_\odot = 6000\text{ K}$, calculate the radius of this supergiant in terms of the Solar Radius ($R_\odot$).

Solution:

Step 1: Substitute the given ratios into the scaling equation.
$$10^5 = \left(\frac{R}{R_\odot}\right)^2 \left(\frac{3000}{6000}\right)^4$$

Step 2: Simplify the temperature ratio.
$$\left(\frac{3000}{6000}\right)^4 = \left(\frac{1}{2}\right)^4 = \frac{1}{16}$$

Step 3: Solve for the Radius ratio.
$$10^5 = \left(\frac{R}{R_\odot}\right)^2 \times \frac{1}{16}$$ $$\left(\frac{R}{R_\odot}\right)^2 = 16 \times 10^5 = 1.6 \times 10^6$$ $$\frac{R}{R_\odot} = \sqrt{1.6 \times 10^6} \approx \mathbf{1265}$$ Conclusion: Despite being twice as cold as the Sun, the star outputs 100,000 times more energy because its radius is over 1,200 times larger!