# A question about star's luminosity, temperature and mass.

## Main Question or Discussion Point

My question is : if two stars have the same luminosity and temperature, do they have to be at the same mass and size?

## Answers and Replies

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mgb_phys
Homework Helper
Typically yes - stars are pretty much blackbodies so their luminosity is a function of temperature and size.

edit - Although that would only be absolutely true for luminosity in the same band - in theory it would be possible for a small hot star to put out the same total energy as a large cool star.

Except for metallicity effects, which can cause stars of the same luminosity and temperature to have drastically different masses.

Which was something I only realized, to my great consternation, about half-way through writing this one paper.

If memory serves me correctly then. Luminosity = 4*pi*r^2*(boltzman constant)*T^4

(where t is temperature and r is the radius of the star).

Whoa! I think it's far too rash to make these sweeping generalizations. The previous posters #2 #3 statements "absolutely true" or "true except for metallicity" are reckless. Stars with the same luminosity could have different sizes and masses for a lot of reasons. I think the age, composition, and the radiative or convective description of the star are paramount. I don't think luminosity is any simple function of a star's mass and size. I'll await an astrophysicist's verdicit on this one.

Whoa! I think it's far too rash to make these sweeping generalizations. The previous posters #2 #3 statements "absolutely true" or "true except for metallicity" are reckless. Stars with the same luminosity could have different sizes and masses for a lot of reasons. I think the age, composition, and the radiative or convective description of the star are paramount. I don't think luminosity is any simple function of a star's mass and size. I'll await an astrophysicist's verdicit on this one.
I have just consulted the Hertzsprung-Russell diagram . It can be an answer for this quest.

Apologies for the late arrival, but now I am in a position of having to ask similar questions to this; I have the task of offering up 'plausible' (if not necessary hyper-accurate) stellar statistics based on only a few initial conditions: mass (and perhaps constituency of that mass in H, He, and metallicities) versus age (how long it has been around.)

This nags me like knowing I know a word that is just beyond the tip of my tongue. I could use all these solar ratio based equations we have derived to 'guess' at a star's features, but it seems to me as though if you know initial mass you know everything. Initial gravity -> initial inward pressure -> required force to initiate fusion -> outward pressure from the energy released by said fusion -> hydrostatic equilibrium -> radius; meanwhile backtracking a bit, that released energy -> radius/surface area -> outward luminosity -> surface temperature. Oversimplistic perhaps, but is there anything missing here?

Apologies for the late arrival, but now I am in a position of having to ask similar questions to this; I have the task of offering up 'plausible' (if not necessary hyper-accurate) stellar statistics based on only a few initial conditions: mass (and perhaps constituency of that mass in H, He, and metallicities) versus age (how long it has been around.)

This nags me like knowing I know a word that is just beyond the tip of my tongue. I could use all these solar ratio based equations we have derived to 'guess' at a star's features, but it seems to me as though if you know initial mass you know everything. Initial gravity -> initial inward pressure -> required force to initiate fusion -> outward pressure from the energy released by said fusion -> hydrostatic equilibrium -> radius; meanwhile backtracking a bit, that released energy -> radius/surface area -> outward luminosity -> surface temperature. Oversimplistic perhaps, but is there anything missing here?
Opacity. Heavier elements in the core means it has to be hotter to fuse and thus fusion goes quicker. That's why stars get brighter as they age on the Main Sequence. Luminosity is usually a simple function of effective temperature and stellar photospheric area - but increased opacity in very cool stars changes that simple relation.

Chronos
Gold Member
The fact stars can be binned in the H-R diagram suggests common denominators - such as mass and elemental composition. Low mass, high luminosity stars and high mass, low luminoisty stars are uncommon.

Opacity. Heavier elements in the core means it has to be hotter to fuse and thus fusion goes quicker. That's why stars get brighter as they age on the Main Sequence. Luminosity is usually a simple function of effective temperature and stellar photospheric area - but increased opacity in very cool stars changes that simple relation.
Makes sense; I shall investigate further. May have more questions after the fact =)

The fact stars can be binned in the H-R diagram suggests common denominators - such as mass and elemental composition. Low mass, high luminosity stars and high mass, low luminoisty stars are uncommon.
Oh, I believe it; that's why I was surprised not to find a 'stellar' equation that could simply equate reasonably predictable stellar 'characteristics' based purely on initial mass and current age (the two details I will know at the outset of my programming.)

Aside from opacity, would steadily decreasing mass (through dissemination of solar wind, energy, CMEs, etc) also technically increase a star's mass as it ages during its main sequence? Less gravity = less inward pressure = new hydrostatic equilibrium upwards = wider radius = more surface area to produce radiance = more luminosity. I realize that the difference might be slim, but can it be a factor? I've ready the sun is 20% brighter today than during its infancy.

Staff Emeritus
Whoa!

If two stars have the same temperature and luminosity, they *must* have the same size. That's a consequence of stars being blackbodies, and has nothing to do with any energy generating mechanism.

They do not have to have the exact same mass, although as a practical matter, it will be fairly close.

Sure, but that's observing only the consequences of the mechanics going on rather than addressing them directly. If not for the energy inside producing resistance to a mass's overall gravity, you couldn't have a hydrostatic equilibrium determining said mass's radius at any given point in time.

i think it's not always there, same luminosity and temperature does not make em of same size there are many stars in the universe which nearly as hot and as bright as the sun (the super red giants) but they are a way big than sun . and also there are starts who are as bright and luminous as proxima centauri but they are big in size . but if you mentioned the age factor also then it can be thought of

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Staff Emeritus
Rishavutkarsh, I am sorry, but none of that is true. The luminosity of a star is determined by its radius and temperature.

Rishavutkarsh, I am sorry, but none of that is true. The luminosity of a star is determined by its radius and temperature.
betelguese is brighter than sun ? sure it isn't it's just as luminous as proxima centauri but if the stages of life of two stars are also included then things can be said this way right?

Staff Emeritus
betelguese is brighter than sun ?
Yes. Technically, it's more luminous. The sun is brighter, but that's because it is closer.

Rishavutkarsh, I am sorry, but none of that is true. The luminosity of a star is determined by its radius and temperature.
And the opacity. In brown dwarfs the formation of dust and clouds obscures the purer black body output of very hot stars. In white dwarfs the thermal resistance of the crust means the star can have an average internal temperature much, much higher than its photosphere. Effective temperature is the equivalent purely blackbody temperature for a given luminosity and photospheric area and is usually pretty close to the spectral temperature. But deviations are what make astronomy interesting/challenging.

Yes. Technically, it's more luminous. The sun is brighter, but that's because it is closer.
Sure; BG is more luminous but due to its bloated dimensions, that is spread out so much more dramatically than either the sun or proxima centauri... you average down (for lack of a better term) to a lower, redder, cooler temperature despite the greater output.

I am curious... I've ready that at outset, a star born here in this galaxy and at this time is going to have X percent hydrogen and Y percent helium and trace amounts of heavier castoffs. When all that aggregates to the point of fusion, (assuming a modest sized star) we speak of it as if it is only fusion hydrogen. Is that technically true? Is there no helium being fused at all until it reaches the end of the main sequence? My mind wants to default to believing that this is occurring in a gradient of a sort, rather than sudden flips of a switch where internal shells of alternately fusing substances materialize.

Sure; BG is more luminous but due to its bloated dimensions, that is spread out so much more dramatically than either the sun or proxima centauri... you average down (for lack of a better term) to a lower, redder, cooler temperature despite the greater output.

I am curious... I've ready that at outset, a star born here in this galaxy and at this time is going to have X percent hydrogen and Y percent helium and trace amounts of heavier castoffs. When all that aggregates to the point of fusion, (assuming a modest sized star) we speak of it as if it is only fusion hydrogen. Is that technically true? Is there no helium being fused at all until it reaches the end of the main sequence? My mind wants to default to believing that this is occurring in a gradient of a sort, rather than sudden flips of a switch where internal shells of alternately fusing substances materialize.
Each "step" has radically higher temperatures and densities at which its probability of occurring rises sufficiently to actually occur, plus each successive set of reactions involves heavier elements that can sink deeper - the heavier fusion "ash" of one reaction becomes the fuel of the next layer downwards because of that.

Each "step" has radically higher temperatures and densities at which its probability of occurring rises sufficiently to actually occur, plus each successive set of reactions involves heavier elements that can sink deeper - the heavier fusion "ash" of one reaction becomes the fuel of the next layer downwards because of that.
Of course; I'm familiar with the concept of shells of fusing different elements... what I wasn't as certain about is what is there that prevents a little helium from being fused along the way prior to the formation of the first helium core? I have some difficulty perceiving the mechanisms by which, SNAP, all of a sudden a helium core spontaneously exists.

So the heavier material sinks; that makes sense... it accumulates... does it reach, maybe, some critical mass where it begins to fuse? It just seems to me that at the center of a star, given higher temps and pressures, some of it would be fusing prior to the formation of a shell.

We should all go dissect a star. =) I don't think the natives will mind.

Of course; I'm familiar with the concept of shells of fusing different elements... what I wasn't as certain about is what is there that prevents a little helium from being fused along the way prior to the formation of the first helium core? I have some difficulty perceiving the mechanisms by which, SNAP, all of a sudden a helium core spontaneously exists.
Hello Jet, I've been reading along and I think I have a few answers. First, helium most certainly burns in the sun; however, the core is not hot enough to support large amounts of this cycle. I believe the p-p chain is responsible for 98% of the suns energy while the CNO is second with ~1.5%. Here is my source: http://neutrino.aquaphoenix.com/un-esa/sun/sun-chapter4.html

Hello Jet, I've been reading along and I think I have a few answers. First, helium most certainly burns in the sun; however, the core is not hot enough to support large amounts of this cycle. I believe the p-p chain is responsible for 98% of the suns energy while the CNO is second with ~1.5%. Here is my source: http://neutrino.aquaphoenix.com/un-esa/sun/sun-chapter4.html
Hah! Much appreciated! I shall investigate.

Hello Jet, I've been reading along and I think I have a few answers. First, helium most certainly burns in the sun; however, the core is not hot enough to support large amounts of this cycle. I believe the p-p chain is responsible for 98% of the suns energy while the CNO is second with ~1.5%. Here is my source: http://neutrino.aquaphoenix.com/un-esa/sun/sun-chapter4.html
Unfortunately the very first line on that source is not correct.

The Sun shines because of the process of fusion....
The sun does NOT shine because of fusion. The sun shines because it is hot. Fusion is a by-product and replenishes lost energy so that the sun may shine for a long time, but fusion doesn't cause the sun to shine.

I'm suspicious of the rest of the page as well... Most of the references are 10-20 years old. I think qraal's answer covered it. The temp and pressure must reach a critical value before helium fusion starts, and if I remember right this is very much a sudden process, hence the 'helium flash.'

Some of the more experienced members might be able to correct me though, if I have misunderstood this concept.