91Ó°ÊÓ

Rotational broadening is a phenomenon that occurs when a star spins on its axis. As the star rotates, the light emitted from its surface experiences a Doppler shift, causing the absorption lines to broaden in the star's spectrum. This effect is especially noticeable when observing a star like the one in our exercise where the equatorial velocity is given as 100 km/s. To determine how much the spectrum is broadened, we use the Doppler shift formula:

• \( \Delta \lambda = \frac{\lambda_0 v}{c} \)

Here, \(\lambda_0\) is the rest wavelength (500 nm), \(v\) is the equatorial velocity (100,000 m/s), and \(c\) is the speed of light (\(3 \times 10^8\) m/s). The result of this calculation gives the full width at half maximum (FWHM), a measure of the broadening effect.
In essence, rotational broadening gives us insight into how quickly a star is spinning, which can affect its shape and surface dynamics.

Thermal broadening occurs due to the random motion of atoms or molecules in a star, caused by thermal energy. As these particles move, they also cause a Doppler shift that affects the absorption lines, broadening them. The higher the temperature, the greater the motion of particles, and consequently, the broader the absorption lines become.

• We use the formula:\( \Delta \lambda = \lambda_0 \sqrt{\frac{2 k T}{m c^2}} \)

Here, \(k\) is the Boltzmann constant, \(T\) is the star's temperature (8000 K), \(m\) is the atomic mass of magnesium in kg, and \(c\) is the speed of light.
To correctly use the formula, we first convert the atomic mass of magnesium from grams to kilograms using Avogadro’s number. This step ensures that the mass aligns with other international system units in our equation, aiding in accurate computation of thermal broadening.

The Doppler effect is a critical concept in stellar spectroscopy, influencing both rotational and thermal broadening. It describes how the frequency or wavelength of light changes due to the relative motion between an observer and the source of the light. When applied to stars:

• If a star or its parts are moving towards us, wavelengths shorten (a blue shift).
• If moving away, wavelengths lengthen (a red shift).

In the context of rotational broadening, different parts of the star surface move towards or away from us due to rotation, creating broader spectral lines. In thermal broadening, it is the random motion due to heat that causes the effect.
The Doppler effect allows us to measure velocities and understand dynamics within stars, providing insight into their structure and behavior.

FWHM, or Full Width at Half Maximum, is a measurement of width of spectral lines at half their maximum intensity. It provides a quantitative measure of how broad a spectral line is, giving insights into the physical processes affecting it. In the context of our exercise:

• For rotational broadening, we calculated the FWHM using the Doppler formula related to equatorial velocity.
• For thermal broadening, FWHM is derived using temperature and particle mass in the thermal broadening formula.

These calculated widths allow us to compare the influence of different broadening mechanisms and determine which is dominant.
Understanding FWHM is crucial in spectroscopy as it relates to the precision of measurements, affecting how we interpret the physical conditions of distant stars.