How Does Age, Mass, and Magnitude Mathematically Relate in Star Clusters?

[randa177]

I've been searching inliterature/trying to solve on my own this dilemma:

What is the mathematical relation that describes the age, mass and apparent magnitude (V) of a star cluster?

I know the mass luminosity relation L=M³
I also know that m = - 2.5 log L
And we can get the age from the HR diagram... but how does it mathematically relate to the magnitude (V) and mass... any idea?

(The reason I am asking this is that the modern stellar synthesis models can create the plot of V vs log age for different masses, but what is the relation governing these factors? )

 

[randa177]

Does my question make sense?

 

[harcel]

Yes, it does. At least most of it. The first relation you quote (L~M^3) is wrong, it holds for stars, not for star clusters. For a star cluster (of sufficiently high mass, higher than about 1000 Msun) the mass and luminosity are linearly proportional to each other (makes sense right? N times more stars, N times more light).

We can get the age from the HR diagram? I presume you mean you can estimate the age from the HR diagram main-sequence turn-off point. That is true. But if you have all stars in an HR diagram too, you might also have its V magnitude (unless it's in bolometric luminosities).

To get the plot you are referring you need to convolve all spectra of all stars (or the composite spectrum of all stars) with the V-band filter profile and compare that to a reference spectrum that belongs to the magnitude system (flat for AB magnitudes, Vega's spectrum for vegamags and so on).

Let me know if you need more info!

 

[randa177]

Yes, it does. At least most of it. The first relation you quote (L~M^3) is wrong, it holds for stars, not for star clusters. For a star cluster (of sufficiently high mass, higher than about 1000 Msun) the mass and luminosity are linearly proportional to each other (makes sense right? N times more stars, N times more light).

We can get the age from the HR diagram? I presume you mean you can estimate the age from the HR diagram main-sequence turn-off point. That is true. But if you have all stars in an HR diagram too, you might also have its V magnitude (unless it's in bolometric luminosities).

To get the plot you are referring you need to convolve all spectra of all stars (or the composite spectrum of all stars) with the V-band filter profile and compare that to a reference spectrum that belongs to the magnitude system (flat for AB magnitudes, Vega's spectrum for vegamags and so on).

Let me know if you need more info!

It took me a while to reply back, but I am still struggling with this question. The problem si that I don't have the spectra of all stars (I actuall don't have any spectrum of any star in the cluster). Is there another way to do it?

 

[pmacias]

The mathematical relation that describes the age, mass, and apparent magnitude (V) of a star cluster is known as the mass-luminosity-age relation. This relation takes into account the mass-luminosity relation, which states that the luminosity of a star is proportional to its mass raised to the third power (L ∝ M^3). It also takes into account the relationship between apparent magnitude and luminosity, which is given by m = -2.5 log L.

The age factor is determined by using the Hertzsprung-Russell (HR) diagram, which plots the luminosity and temperature of stars. By comparing the location of a star on the HR diagram to theoretical models, the age of the star can be estimated.

So, the mathematical relation can be written as:

V = -2.5 log (M^3) + b log(age)

Where b is a constant that takes into account the relationship between apparent magnitude and luminosity. This equation shows that the apparent magnitude of a star cluster is dependent on its mass and age. As the mass increases, the luminosity and thus the magnitude increases. Similarly, as the age increases, the luminosity decreases and the magnitude also decreases.

It is important to note that this relation is based on theoretical models and may not accurately represent the actual properties of a star cluster. Additionally, the constant b may vary depending on the composition and evolutionary stage of the stars in the cluster.

In conclusion, the mass-luminosity-age relation provides a mathematical description of the relationship between the age, mass, and apparent magnitude of a star cluster. It is a useful tool for understanding the properties of stars and their evolution.