E5-4. Stellar Parallax and Determination of Stellar Radii

1. Stellar Parallax

Stellar Parallax is a geometric method used to measure the distance to relatively nearby stars. As the Earth orbits the Sun, nearby stars appear to shift slightly back and forth against the stationary background of incredibly distant stars.

Stellar Parallax Geometry Sun Earth (July) Earth (Jan) Target Star Distance (d) 1 AU p Distant Background Stars Apparent position (Jan) Apparent position (July)
  • The parallax angle ($p$) is defined as half the total angular shift observed over a 6-month period.
  • If the parallax angle $p$ is measured in arcseconds ($1'' = \dfdrac{1}{3600}$ of a degree), the distance $d$ to the star in parsecs (pc) is given by the simple formula:
    $$d \ (\text{in pc}) = \frac{1}{p \ (\text{in arcseconds})}$$
  • A parsec is approximately $3.26\text{ light-years}$ or $3.09 \times 10^{16}\text{ m}$.
  • Because atmospheric distortion limits Earth-based telescope resolution to about $0.01''$, this method is traditionally only viable for stars up to $\approx 100\text{ pc}$ away. Space-based telescopes (like Gaia) can push this limit to thousands of parsecs.

2. Apparent Brightness and the Stefan-Boltzmann Law

Apparent Brightness ($b$) vs. Luminosity ($L$)

  • Luminosity ($L$): The total absolute power emitted by a star in all directions (measured in Watts). It is an intrinsic property of the star.
  • Apparent Brightness ($b$): The power received per unit area at Earth (measured in $\text{W m}^{-2}$). Due to the inverse-square law of spherical expansion, brightness drops off rapidly with distance $d$:
    $$b = \frac{L}{4\pi d^2}$$

Determining Stellar Radii

If we treat a star as a perfect spherical black body radiator, the Stefan-Boltzmann Law states that its Luminosity depends on its surface area ($A = 4\pi R^2$) and its absolute temperature ($T$):

$$L = \sigma A T^4 \implies L = 4\pi R^2 \sigma T^4$$

Where $\sigma = 5.67 \times 10^{-8}\text{ W m}^{-2}\text{ K}^{-4}$ (the Stefan-Boltzmann constant). By determining $L$ from apparent brightness and distance, and $T$ from Wien's law, astronomers can calculate the exact physical radius ($R$) of a star without ever physically traveling to it!

3. Examples

Example 1: Parallax Limits & Distance Conversions

Problem: The star Alpha Centauri exhibits a parallax angle of $0.742\text{ arcseconds}$.
(a) Calculate the distance to Alpha Centauri in parsecs and light-years ($1\text{ pc} = 3.26\text{ ly}$).
(b) Explain mathematically why the parallax method fails for stars located in the Andromeda Galaxy (2.5 million light-years away).

Solution:

Part (a): Distance Calculation
$$d = \frac{1}{p} = \frac{1}{0.742''} = \mathbf{1.348\text{ pc}}$$ Convert to Light-years: $$d = 1.348\text{ pc} \times 3.26\text{ ly/pc} \approx \mathbf{4.39\text{ light-years}}$$

Part (b): Method Limitations
For Andromeda ($d = 2,500,000 / 3.26 = 766,871\text{ pc}$), the parallax angle would be: $$p = \frac{1}{d} = \frac{1}{766,871} \approx 0.0000013''$$ This angle is vastly smaller than the optical resolution limit of any telescope currently in existence, making the shift utterly undetectable against the background noise.

Example 2: Multi-step Stellar Radii Calculation (Ultimate Exam Synthesis)

Problem: Betelgeuse is a red supergiant star. A telescope measures its peak emission wavelength to be $\lambda_{max} = 850\text{ nm}$. Its apparent brightness as observed from Earth is $b = 1.0 \times 10^{-7}\text{ W m}^{-2}$, and its distance is calculated via satellite parallax to be $d = 200\text{ parsecs}$.
Calculate the physical radius of Betelgeuse in meters, and then express it as a ratio of the Solar Radius ($r_\odot = 6.95 \times 10^8\text{ m}$).

Solution:

Step 1: Calculate Surface Temperature (Wien's Law)
$$T = \frac{2.9 \times 10^{-3}}{\lambda_{max}} = \frac{2.9 \times 10^{-3}}{850 \times 10^{-9}} = \mathbf{3412\text{ K}}$$

Step 2: Convert Distance to Meters
$$d = 200\text{ pc} \times (3.09 \times 10^{16}\text{ m/pc}) = \mathbf{6.18 \times 10^{18}\text{ m}}$$

Step 3: Calculate Total Luminosity (Inverse Square Law)
$$L = b \times 4\pi d^2 = (1.0 \times 10^{-7}) \times 4 \times \pi \times (6.18 \times 10^{18})^2 = \mathbf{4.80 \times 10^{31}\text{ W}}$$

Step 4: Calculate the Stellar Radius (Stefan-Boltzmann Law)
$$L = 4\pi R^2 \sigma T^4 \implies R = \sqrt{\frac{L}{4\pi \sigma T^4}}$$ Calculate the denominator: $4 \times \pi \times (5.67 \times 10^{-8}) \times (3412)^4 \approx 9.64 \times 10^{10}$ $$R = \sqrt{\frac{4.80 \times 10^{31}}{9.64 \times 10^{10}}} = \sqrt{4.98 \times 10^{20}} = \mathbf{7.06 \times 10^{10}\text{ m}}$$

Step 5: Compare to the Sun
$$\text{Ratio} = \frac{R}{r_\odot} = \frac{7.06 \times 10^{10}}{6.95 \times 10^8} \approx \mathbf{1016}$$ Betelgeuse has a radius approximately 1,000 times larger than our Sun! If placed in our solar system, its surface would swallow Jupiter.