Stellar Parallax, Luminosity, Spectral Analysis, and Astronomical Unit Conversions
- You are standing on a quiet beach, watching a distant ship sail across the horizon.
- As you take a few steps along the shore, the ship appears to shift slightly against the backdrop of the sky.
- This apparent shift, caused by viewing the ship from two different positions, mirrors how astronomers measure the distances to nearby stars using stellar parallax.
Stellar Parallax: Measuring the Distance to Stars
Stellar Parallax
Stellar parallax refers to the apparent shift in the position of a nearby star against the background of distant stars when observed from two opposite points in Earth’s orbit around the Sun, six months apart.
This phenomenon is one of the most fundamental methods for determining stellar distances.
The Parallax Angle and Distance Formula
Parallax Angle
The parallax angle $p$ is defined as half the total angular shift of the star.
The smaller the parallax angle, the farther away the star is.
The relationship between the parallax angle and the distance ($d$) to the star is expressed as:
$$d \, (\text{parsecs}) = \frac{1}{p \, (\text{arc-seconds})}$$
where:
- $d$ is the distance to the star in parsecs ($\mathrm{pc}$),
- $p$ is the parallax angle in arc-seconds.
Suppose a star has a parallax angle of $0.1 \, \text{arc-seconds}$. Using the formula:
$$d = \frac{1}{p} = \frac{1}{0.1} = 10 \, \text{parsecs}$$
NoteOne parsec (pc) is approximately $3.09 \times 10^{16} \, \text{m}$ or $3.26 \, \text{light years (ly)}$.
Limitations of Parallax
- While powerful, the parallax method is limited to nearby stars.
- For distant stars, the parallax angles become so small that they are difficult to measure accurately.
Ground-based telescopes can measure parallaxes for stars up to about $100 \, \text{pc}$ away, whereas space-based observatories like Gaia extend this range to approximately $3000 \, \text{pc}$.
Self reviewUsing the formula $d = \frac{1}{p}$, calculate the distance to a star with a parallax angle of $0.05 \, \text{arc-seconds}$.
Luminosity and Temperature: The Stefan-Boltzmann Law
Luminosity
Luminosity measures the amount of radiated electromagnetic energy per unit time.
- Stars emit light and heat, and their luminosity ($L$), the total energy radiated per second, depends on their surface temperature ($T$) and size (radius, $R$).
- This relationship is captured by the Stefan-Boltzmann Law: $$L = 4\pi R^2 \sigma T^4$$ where:
- $L$ is the star’s luminosity (in watts),
- $R$ is the radius of the star (in meters),
- $T$ is the surface temperature (in kelvin),
- $\sigma = 5.67 \times 10^{-8} \, \text{W m}^{-2}\ \text{K}^{-4}$ is the Stefan-Boltzmann constant.
