# Inferring Properties of Stars from Masses and Radii

I have two questions related to inferring properties of stars from their masses and radii.

1. What properties of a star's spectrum could we deduce? In particular, do all stars emit like black bodies? If so, I'm guessing that one can calculate the star's effective temperature and use Wien's displacement law to find the peak of the black body spectrum.
2. How would the mass and radius influence Type 1 X-ray bursts?

All evolved stars do emit like black bodies, but this is not true for young stellar objects which may exhibit a large infrared excess.

Thus if you consider, for example, a main sequence star (which are the most common stars) and you measure its mass $$M$$ and radius $$R$$, you can indeed infer its spectral properties. Firstly, let us write the equation for the luminosity $$L$$ of the star:

$$L = 4 \pi R ^ 2 F$$

Where $$F$$ is the spectral flux density at the surface of the object.

If you assume that the emission can be described by the black body model, then you have:

$$L = 4 \pi R ^ 2 \sigma T ^ 4$$

Where $$\sigma$$ is the Stefan-Boltzmann constant and $$T$$ the temperature.

Now you need a mass-luminosity function to relate the luminosity $$L$$ to the mass $$M$$ that you have measured. This relation can be derived theoretically, by thermodynamic considerations, and yields this kind of equation:

$$L \propto M ^ a$$

In the case of main sequence stars, this relation is well known and the value $$a = 3,5$$ is commonly used.

Thus, using the previous equation, it is possible to determine the surface temperature of our main sequence star:

$$T = \left( \dfrac{M ^ {3,5}}{4 \pi R ^ 2 \sigma }\right) ^ {1/4}$$

Then, as you said, you can benefit from Wien's displacement law to find the wavelength $$\lambda_{max}$$ of the emission peak in the black body spectrum, given the temperature $$T$$ that you just found, using:

$$\lambda_{max} = \dfrac{b}{T}$$

By doing so you have determined the spectral type of the star only from $$M$$ and $$R$$. Be warned however that this depends on the choice of the mass-luminosity relation.

Note: Additionnaly, having determined the luminosity $$L$$ and temperature $$T$$ of the star would allow you to place it on a Hertzsprung–Russell diagram.