E.5.3 Measuring stellar properties

Stellar Parallax, Luminosity, Spectral Analysis, and Astronomical Unit Conversions

1. You are standing on a quiet beach, watching a distant ship sail across the horizon.
2. As you take a few steps along the shore, the ship appears to shift slightly against the backdrop of the sky.
3. 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

This phenomenon is one of the most fundamental methods for determining stellar distances.

The Parallax Angle and Distance Formula

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.

Limitations of Parallax

1. While powerful, the parallax method is limited to nearby stars.
2. For distant stars, the parallax angles become so small that they are difficult to measure accurately.

Luminosity and Temperature: The Stefan-Boltzmann Law

1. Stars emit light and heat, and their luminosity ($L$)—the total energy radiated per second—depends on their surface temperature ($T$) and size (radius, $R$).
2. 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),
• $\sigma =5.67\times {10}^{-8}{\text{Wm}}^{-2}{\text{K}}^{-4}$ is the Stefan-Boltzmann constant.

Spectral Analysis: Unlocking a Star’s Secrets

1. The light from a star carries valuable information about its physical properties, including its chemical composition and temperature.
2. This information is uncovered through spectral analysis, the study of a star’s spectrum.

Chemical Composition

1. As light passes through a star’s outer layers, certain wavelengths are absorbed by elements in its atmosphere.
2. These absorbed wavelengths appear as dark lines in the star’s spectrum, called absorption lines.

Temperature Determination

The peak wavelength (${\lambda }_{\text{max}}$) of a star’s emitted light is related to its surface temperature by Wien’s Law:

$$T=\frac{2.9\times {10}^{-3}}{{\lambda }_{\text{max}}}$$

where:

• $T$ is the temperature in kelvin,
• ${\lambda }_{\text{max}}$ is the peak wavelength in meters.

Conversions Between Astronomical Units

1. Astronomy often involves vast distances expressed in units like astronomical units (AU), light years (ly), and parsecs (pc).
2. Converting between these units is essential for accurate calculations.

Key Relationships

1. $1\text{AU}=1.496\times {10}^{11}\text{m}$ (the average Earth-Sun distance).
2. $1\text{ly}=9.461\times {10}^{15}\text{m}$ (the distance light travels in one year).
3. $1\text{pc}=3.09\times {10}^{16}\text{m}=3.26\text{ly}$.

Reflection