# Star Radius & Mass from Spectral Class, B-V, Luminosity

• I
• gunhed508
In summary: So, if someone wanted to try their hand at calculating Surface Temperatures using B-V colors, and wasn't confident in their math, it might be a better idea to use the UNI equation.In summary, the data set includes the spectral class, B-V color index, and luminosity of stars. The spectral codes outside of the typical OBAFGKM set, such as D, C, S, R, W, N, and P, determine the temperature. If you have the luminosity and the temperature, you have the radius.
gunhed508
I have a data set of 120k star systems that I'd like to import into a project, and, while it has a lot of useful infomation, I'd like to display these stars in a visual fashion. This means that I need to figure out the radius, when zoomed into the star system, and its mass, to simulate objects in orbit. If I have the radius, I can appoximate its mass. I have a basic idea of how to color them.

The data set I've chosen gives me the Spectral Class, the B-V Color Index, and the Luminosity of each star, and I'm hoping that's enough to plug into some math out there to approximate the radius and/or mass. When I checked out this data, I found some weird spectral codes outside of the typical, established OBAFGKM set, such as D, C, S, R, W, N, and P. This code will basically determine the Temperature, as I understand it, thus determining the Radius, but I'm not sure where these fit in. "D" is a white dwarf, I think, for example, and "C" is a Carbon star, but I don't have any temperature info on those or the others.

My main question is this: how do I calculate the stars' radii and/or mass from the data I have?

The data set I'm looking at is here:
http://astronexus.com/node/34
The description of the data headers is here:

This is the breakdown of spectral classes:
32182 K
25605 F
22798 G
18697 A
10413 B
5831 M
3050 <no data>
282 D
264 O
162 C
102 S
89 R
74 W
61 N
4 P

If you have the luminosity and the temperature, you have the radius, viz

L = σT44πR2

Why don't you first work on the majority of stars whose spectral classes you understand before proceeding to the more rare cases (which are defined here https://en.wikipedia.org/wiki/Stellar_classification)

gunhed508
DrSteve said:
If you have the luminosity and the temperature, you have the radius, viz

L = σT44πR2

Why don't you first work on the majority of stars whose spectral classes you understand before proceeding to the more rare cases (which are defined here https://en.wikipedia.org/wiki/Stellar_classification)

Thank you for the equation and the great link - looks like it has a nice table there for the common spectral classes. I noticed there's a numeric range that can follow afterwards: 0 being hottest, and 9 the coolest. Is this a linear correlation? For example, for a G class star, the range is 5200-6000. Should I expect a G5 to be something like 5644 K?

It's unfortunate that I'm only seeing useful data for Main Sequence stars. As for the oddballs - this project has a couple of goals: First, to use as much hard-core science as I can. Second, to present a "journey through the stars," where the occasional rare, oddball star is actually quite desireable. Who wouldn't want to behold the glory of a red supergiant?

DrSteve
I just found something interesting - there's an article about Finding the Radius of a Star, and in it is a nice table relating the B-V color index to the star's Temperature. Since my data set includes the B-V values of each star, why can I not derive the temperature directly from this value? What is the actual equation that generated that table? If I had that math, I could possibly simplify my work. Can anyone point me to this math?

Wow - I guess there aren't many Astrophysicists reading this forum. :( I seemed to have found some answers on my own without resorting to cornering a very busy professor at my local college. :) For those interested in this subject, particularly what's involved in rendering the approximate color, size, and luminosity of local stars, I've done a lot of the legwork for you!

In my searching, I found a couple of potential equations to estimate a star's Surface Temperature from its BV color index. I used good-old MS Excel to plot the simplified table from my previous post to compare it to the two equations.

The first is from Yahoo! Answers, of all places, from a person at UTK who left this post. The equation follows:
If B-V > -0.0413, then
T = 10^{ [14.551 - (B-V)] / 3.684 }

If B-V < -0.0413, then
T = 10^{ 4.945 - sqrt[ 1.087353 + 2.906977 (B-V) ] }

The second post was from an individual at UNI, who had created an online http://www.uni.edu/morgans/stars/b_v.html to perform the calculation, which looks like a least-squares curve fit for a table using 6 or 7 data points. I gleaned this equation (using the provided table of constants) from the javascript code:
C1=3.9791451067;
C2=-0.6544992269;
C3=1.7406900424;
C4=-4.6088151541;
C5=6.7925997799;
C6=-5.3969098913;
C7=2.1929703765;
C8=-0.3594957393;
logt=C1+C2*bmv+C3*pow(bmv,2)+C4*pow(bmv,3)+C5*pow(bmv,4)+C6*pow(bmv,5)+C7*pow(bmv,6)+C8*pow(bmv,7);
t=pow(10,logt);

As I said, I plotted both equations over the original table, and compared results. Despite the UTK method requiring an inequality, I found that its curve seemed to fit better. The UNI method fit OK, but at the endpoints, the curve seemed to become more exaggerated and less reliable.

Once I knew the temperature, I could calculate the Radius of the star using equations derived from the Boltzmann Law:
R/Rsol = (Tsol/T)^2 * SQRT(L/Lsol)

Since I'm using the HYG data set, the luminosity is already expressed in ratio to Sol, so it made at least one thing easy. Also, I attempted to use the spetrum data from the set (where possible) to infer the temperature, and, thus the radius using some work already curated by Landon Curt Noll. Some spectrum data was imcomplete, so I made my best guess. Those gaps in the data where I had no spectrum data, I used the B-V color index to calculate the temperature. Those stars with no spectrum or color index were dropped from my data set.

The result, thusfar, is this:
http://www.marsgate.net/stars/starmap.html
Please note this is a work in progress, and will only function smoothly on high-end computers; I doubt a tablet could handle it. Also, it's not designed to work with Internet Explorer, as I found it lacking in many respects.

## 1. How is the radius of a star determined from its spectral class?

The radius of a star is determined from its spectral class by using the Stefan-Boltzmann law, which relates the temperature and luminosity of a star to its radius. By knowing the temperature of a star based on its spectral class, scientists can calculate its radius using this law.

## 2. How does B-V color index help determine the mass of a star?

The B-V color index, also known as the color temperature index, is a measure of the color of a star, with hotter stars having a lower index and cooler stars having a higher index. By knowing the B-V index and temperature of a star, scientists can use the Hertzsprung-Russell diagram to estimate the star's mass based on its position on the graph.

## 3. Can the luminosity of a star be used to determine its mass and radius?

Yes, the luminosity of a star can be used to determine its mass and radius. By knowing the luminosity of a star, scientists can use the mass-luminosity relationship, which states that more massive stars have a higher luminosity, to estimate the star's mass. Then, by using the Stefan-Boltzmann law, the radius of the star can be calculated.

## 4. How accurate are the calculations for determining a star's radius and mass using spectral class, B-V index, and luminosity?

The calculations for determining a star's radius and mass using spectral class, B-V index, and luminosity are relatively accurate. However, there can be some uncertainties due to factors such as the presence of a companion star, stellar variability, and errors in measurements. Therefore, the estimated values should be considered as approximations rather than precise measurements.

## 5. Can these methods be used for all types of stars?

These methods can be used for most types of stars, including main sequence stars, giant stars, and white dwarfs. However, for highly evolved stars such as red giants and supergiants, additional factors may need to be taken into account, and the calculations may not be as accurate due to their complex structures and variability.

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