# What is the definition of the "stellar angular diameter" in stellar astronomy?

(Following the definitions here: Teff and log g Determinations)

What is the "stellar angular diameter", as measured by astronomers specializing in stellar astrophysics?

Using the Stefan-Boltzmann law, the effective temperature of a star, $$\text{T}_\text{eff}$$, is defined as:

$$\sigma\text{T}_\text{eff}^4=\int^{\infty}_0 F_{\nu}d\nu=\text{F}_{*}=\frac{\text{L}}{4\pi R^2}$$

where luminosity is denoted $$\text{L}$$ and total radiant power per unit area at the stellar surface is $$\text{F}_{*}$$. Stellar luminosity is defined as $$\text{L}_{*}=4\pi R^2_{*}\sigma\text{T}_{*}^4$$.

The integrated radiative flux at the stellar surface is $$\text{F}=\int^{\infty}_0F_{\nu}d\nu$$.

Denoting the total observed flux at earth to be $$f_{\oplus}$$, the total flux of the star is determined by

$$\text{F}_{*}=\frac{\theta^2}{4}f_{\oplus}$$.

The stellar angular diameter is $$\theta$$. What is this?

Apparently, $$\theta$$ is measured using speckle photometry, interferometry, and lunar occultations, and indirectly measured from eclipsing binary systems of known distances.

• I'm guessing it is the diameter of the star as measured from a telescope from Earth? Aug 28, 2015 at 23:35
• What in the Wikipedia article on angular diameter doesn't answer your question? Sep 3, 2015 at 7:59
• @Warrick Yes, why is the quantity squared and divided by 4? Sep 4, 2015 at 1:46

## 1 Answer

The angular diameter (in radians) is the physical diameter of the star's photosphere divided by the distance to the star. i.e. $$\theta = \frac{2R}{d}\ ,$$ where $$R$$ is the stellar radius and $$d$$ is the distance to the star.

The flux at the Earth is $$f_\oplus = \frac{4\pi R^2 F_*}{4\pi d^2} = \frac{R^2}{d^2}F_* = \frac{\theta^2}{4}F_*\ .$$

i.e. I think you have your relationship back to front.