Star variable relations. In need of further insight

[Geonaut]

Are the absolute magnitude of stars relevant (calculated) with it's mass and rotational velocity? I understand that the mass of a star has a relation to it's size... regardless of this.. I need to know if mass and rotational velocity combined are used to determine anything for stars. Furthermore I've read that the variables of mass and a 4-vector rotational velocity of a black hole are used to determine it's ergosphere momentum. Does this 4-vector rotational velocity have a mathematical relation to a rotational velocity to the fourth power.. meaning can it be used this way instead? I've been researching for a few days... it's hard to find such specific details.


Assistance would be appreciated, thank you

 

[Geonaut]

I suppose that may have been too many questions... What I really need to know is: Can the temperature of a star be calculated using mass and rotational velocity?



thanks again

 

[Chalnoth]

Well, I don't think rotation velocity has much of anything to do with the temperature of a star. There may be some small effect in terms of the pressure in the interior, but it is going to be quite small indeed.

And a 4-vector has nothing whatsoever to do with a fourth power. Read up on the four-velocity here:
http://en.wikipedia.org/wiki/Four-velocity

Basically, the four-velocity is just a mathematically-useful way of writing down the velocity that is easy to make use of in relativistic calculations. It's just the velocity written down differently is all.

Some of the things that we can learn about stars from observing them are as follows:
1. We can independently observe the temperature, distance, and brightness of many stars. These observations, when combined, can give us the physical size of the star (because temperature gives the amount of radiation emitted per unit area, we can calculate the surface are of the star from the distance and observed brightness).
2. We can obtain the mass of the star if it is in a binary system by measuring the orbit of the other star.

Most often, we can't even measure the rotational velocity of the star.

 

[Geonaut]

hmmm temperature and brightness may be used to determine size... which may be used to determine mass.. I've read that the mass of a star (or it may have been the star's size) may be used to determine it's rotational velocity, they go hand and hand. I assumed that was the result of mass acting as a gravational fuel for acceleration during star formation causing it's inital rotational velocity to correspond.

I attempted to teach myself what the 4-vector quantity was "driving at" math wise. I recall a symbol being used with a power variable such as "x" and that the "x" was based on the number of dimensions involved. The equation for ergospheres was formulated using Einstein methods resulting with a 4 dimensional situation and so I had assumed...? V^4 ?

 

[Chalnoth]

hmmm temperature and brightness may be used to determine size... which may be used to determine mass..

Well, not really. A star with the same mass will have very different temperature/brightness over its lifetime, depending upon what sort of nuclear reactions are going on in its core at any given time. I suppose you can sort of infer its mass, if you make some assumptions about what nuclear reactions are going on (which can be done, with limited accuracy, by correlating the star with others with known mass).

I've read that the mass of a star (or it may have been the star's size) may be used to determine it's rotational velocity, they go hand and hand.

Well, sort of, but only in a probabilistic sense. When a diffuse gas of randomly-assorted matter collapses, chances are it will have some net angular momentum. Because the star is much, much smaller than the matter which collapsed into it, it tends to pick up an appreciable spin. But since this spin came from the random motions of the molecules making up this gas, it can't be predicted very well.

Now, as the star evolves, and it becomes larger or smaller, the spin will change accordingly. But it will still depend upon the rate of spin of that initial bit of matter.

I attempted to teach myself what the 4-vector quantity was "driving at" math wise. I recall a symbol being used with a power variable such as "x" and that the "x" was based on the number of dimensions involved. The equation for ergospheres was formulated using Einstein methods resulting with a 4 dimensional situation and so I had assumed...? V^4 ?

I really have no clue what you're thinking of here.

But the basic idea behind a four-vector is this. Because relativity deals with space-time, a four-vector position was developed which includes one time coordinate and three spatial coordinates. This turns out to be quite convenient because it is possible to convert this position from one observer's coordinates to another's with a simple matrix multiplication. It turns out that you can produce four vectors for other common quantities, such as velocity and energy-momentum, that also transform in this same way.

 

[IsometricPion]

I recall a symbol being used with a power variable such as "x" and that the "x" was based on the number of dimensions involved.

I believe you mean something like this: $$x^{\nu},\nu\in\{1,2,3,4\}$$, i.e.,$$x^{4}$$, etc. This is a form of mathematical notation and is not meant to imply raising the value of a coordinate to some power, rather it is meant to compactly convey how a given relation applies for all coordinates (3 spatial dimensions and 1 temporal). The number 4 as a superscript denotes the time dimension, e.g., $$F^{4}=\frac{dP^{4}}{d\tau}$$ means the time component of the force 4-vector is equal to the derivative of the time component of the momentum 4-vector with respect to the proper time along a given path in space-time. The other three numbers denote the three spatial dimensions (in whatever coordinates they happen to be characterized, e.g., spherical coordinates of $$\rho,\theta,\phi$$). So, $$F^{3}=\frac{dP^{3}}{d\tau}$$ means (in cartesian coordinates where I can, e.g., assign 1 to denote the x-coordinate, 2 the y-coordinate, and 3 the z-coordinate) that the z component of the force four-vector is equal to the derivative of the z component of the momentum 4-vector with respect to the same proper time variable as before.

4-vectors, in general, are just vectors in a four dimensional vector space. Similar to vectors in three dimensions, they can be represented in whatever set of coordinates one wishes as a list of four numbers, e.g., $$(1,0,0,0)$$. In the case of special relativity the only rule that is different from vectors in three dimensions is the way the norm of a vector is defined. The definition is: $$||\vec{F}||=\sqrt{|(F^{1})^{2}+(F^{2})^{2}+(F^{3})^{2}-c^{2}(F^{4})^{2}|}$$ where c is the speed of light, ref. http://en.wikipedia.org/wiki/Minkowski_space" (note that the article uses the more common convention where 0 denotes the time coordinate).

This notation is useful when tensors are involved, especially ones with more than two dimensions (indices) http://en.wikipedia.org/wiki/Abstract_index_notation" .

 

[twofish-quant]

Are the absolute magnitude of stars relevant (calculated) with it's mass and rotational velocity? I understand that the mass of a star has a relation to it's size... regardless of this.. I need to know if mass and rotational velocity combined are used to determine anything for stars.

Not really.

The relation is pretty much

Mass -> nuclear reaction rate -> surface temperature.

Yes you can calculate the surface temperature from the mass, but this more of a "dump the equations into the computer" sort of thing rather than something simple and elegant you can write in four lines.

Black holes are very different.

 

[Geonaut]

Wow, thanks a lot... there's more.

"Deriving the second stellar parameter--diameter--is trickier. To do so, Jeffries compared two different measures of the stars' rotation. First, other astronomers have been able to determine how long the stars take to spin. That's because starspots make the stars brighten and fade as they rotate. Monitoring the star's brightness therefore reveals the rotation period". -Quote from Work Cited: http://kencroswell.com/OrionNebulaDistance.html

A star's temperature to the fourth power may be used to determine the star's luminosity which can be used to determine the star's rotational period.

That's how I saw this anyway which then lead to many questions. I am sorry I wasn't clear with my previous statements. I was busy at the time, and didn't have much time to put much time into the questions.

 

[Chalnoth]

Wow, thanks a lot... there's more.

"Deriving the second stellar parameter--diameter--is trickier. To do so, Jeffries compared two different measures of the stars' rotation. First, other astronomers have been able to determine how long the stars take to spin. That's because starspots make the stars brighten and fade as they rotate. Monitoring the star's brightness therefore reveals the rotation period". -Quote from Work Cited: http://kencroswell.com/OrionNebulaDistance.html

This all is reasonable and interesting, but...

A star's temperature to the fourth power may be used to determine the star's luminosity which can be used to determine the star's rotational period.

Where do you get this from??

 

[Geonaut]

"Other astronomers have measured spectral types for the stars and thereby deduced stellar temperatures. Since a star's luminosity depends on the fourth power of its temperature, a doubling of temperature corresponds to a sixteen-fold increase in luminosity". - Quote of work Cited: http://kencroswell.com/OrionNebulaDistance.html

I'm wondering if I put two and two together accurately, or if I misinterpreted something.

 

[Chalnoth]

"Other astronomers have measured spectral types for the stars and thereby deduced stellar temperatures. Since a star's luminosity depends on the fourth power of its temperature, a doubling of temperature corresponds to a sixteen-fold increase in luminosity". - Quote of work Cited: http://kencroswell.com/OrionNebulaDistance.html

I'm wondering if I put two and two together accurately, or if I misinterpreted something.

Um, the measurement of the stars' luminosities had nothing to do with the measurement of the stars' rotation periods. If you read the article, they estimate the stars' diameter by comparing two different measures of the stars' rotation.

Looking very closely for periodic dips in stellar luminosity gives the rotation period (because these dips are caused by sunspots which persist on the star's surface for some time, rotating with it). Looking very closely at the broadening of spectral lines gives the blueshift/redshift of the edges of the star as it rotates, giving an estimate of its rotation velocity. Combining these two different measures allows for an estimate of the actual diameters of the stars.

Once they combine the actual diameter with the temperature, they get an estimate of the intrinsic luminosity, which can be used as a measure for distance by comparing against the observed luminosity.