The most massive of the four bright stars in the Trapezium...

The most massive of the four bright stars in the Trapezium open cluster of the Orion Nebula is Theta Orionis C. Its spectrumluminosity class is $\mathrm{O} 6 \mathrm{~V}$, its visual apparent magnitude is $m_{\mathrm{v}}=5.13$, and its absolute visual magnitude is $M_{\mathrm{v}}=-3.2$. It is estimated to have a radius of $8 R_{\odot}$, a mass of $40 M_{\odot}$, and its luminosity is $251,000 L_{\odot}$ (much of this at UV wavelengths, since Theta Orionis C has the distinction of having the hottest surface temperature of any star visible to the naked eye). What is the parallax of Theta Orionis C? What is its surface temperature? Given that Theta Orionis C is thought presently to be about 1 million years old, how much longer will it spend on the main sequence?

Key Concepts

Surface Temperature and the Stefan-Boltzmann Law

A star's surface temperature is related to its luminosity and radius by the Stefan-Boltzmann law, which states that the luminosity of a star is proportional to its surface area and the fourth power of its temperature. Combined with the star's spectral classification, this law helps astronomers determine the effective temperature of a star by comparing its observed brightness and energy distribution with that of a blackbody.

Main Sequence Lifetime and Stellar Evolution

The main sequence lifetime of a star refers to the period during which it fuses hydrogen into helium in its core. This phase is largely controlled by the star's mass, with more massive stars consuming their nuclear fuel at a much faster rate than lower-mass stars, resulting in shorter lifetimes on the main sequence. Stellar evolution theory, along with the mass-luminosity relation, is used to estimate how long a star will remain in this stable phase before evolving into later stages.

Stellar Parallax

Stellar parallax is the apparent shift of a nearby star's position relative to distant background stars as observed from different positions of Earth in its orbit around the Sun. This effect is used to calculate the distance to the star, with the parallax angle inversely proportional to the distance; the smaller the angle, the farther the star is from Earth. The measurement is typically expressed in arcseconds, and the distance is given in parsecs.

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