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Ken G
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- Everyone likes an explanation that seems to make sense, sometimes called a "truthy" explanation. A great thing about science is it teaches us to dig past the "truthy", but scientific websites, including introductory astronomy courses, often fail to do this. Consider these spectacularly false explanations for the celebrated "mass luminosity relation" in main sequence stars.
It is widely known that the more massive a main sequence star, the higher its luminosity, which is called the mass-luminosity relation. This relation is so important, many authors and educators cannot resist the temptation to include an explanation for it. While I applaud that sentiment, unfortunately they often settle for a "truthy" explanation that sounds reasonable but is actually no better than something Calvin's dad would say! Often, we hear that higher mass stars have higher gravity and higher core pressure, which is patently false (one of the most important things to understand about main sequence stars is that they all have similar core temperatures, so the more massive ones need to be larger, have lower density, weaker gravity, and lower core pressure!). So that explanation is the opposite of correct, yet it comes up in all of the initial hits one gets when searching the web for introductory sites (the sites that do not give mathematical derivations). For example, consider the following, which seem like authoritative efforts at explaining the relation:
Penn State: https://www.e-education.psu.edu/astro801/content/l7_p3.html
"Since higher mass means a larger gravitational force, higher mass must also mean that higher pressure is required to maintain equilibrium. If you increase the pressure inside a star, the temperature will also increase. So, the cores of massive stars have significantly higher temperatures than the cores of Sun-like stars. At higher temperatures, the nuclear fusion reactions generate energy much faster, so the hotter the core, the more luminous the star."
Open Stax: https://openstax.org/books/astronomy-2e/pages/18-4-the-h-r-diagram
"The most massive stars have the most gravity and can thus compress their centers to the greatest degree. This means they are the hottest inside and the best at generating energy from nuclear reactions deep within. As a result, they shine with the greatest luminosity and have the hottest surface temperatures."
Swinburne University: https://astronomy.swin.edu.au/cosmos/M/Main+Sequence+Lifetime
"Massive stars need higher central temperatures and pressures to support themselves against gravitational collapse, and for this reason, fusion reactions in these stars proceed at a faster rate than in lower mass stars. The result is that massive stars use up their core hydrogen fuel rapidly..."
Astronomy Notes: https://www.astronomynotes.com/starsun/s8.htm
"Massive stars have greater gravitational compression in their cores because of the larger weight of the overlying layers than that found in low-mass stars. The massive stars need greater thermal and radiation pressure pushing outward to balance the greater gravitational compression. The greater thermal pressure is provided by the higher temperatures in the massive star's core than those found in low-mass stars. Massive stars need higher core temperatures to be stable!"
So there you have it, complete consensus, totally wrong. Yet what student could possibly doubt the veracity of such an authoritative consensus? One who understands the core premise of science: be skeptical, think for yourself. If you want the correct reason, look at Wikipedia, or this Princeton course website:
Princeton University: https://www.astro.princeton.edu/~gk/A403/massive.pdf
"It is remarkable that we obtained the mass luminosity relation without any reference to the stellar
energy sources. This can be understood in the following terms. Within our approximation the
opacity of matter is constant (per unit mass), the photons diffuse out at the rate they can, which is
independent on stellar temperature or density. So the heat losses are fixed by the constant opacity. If
there are no nuclear energy sources then the star will be losing energy, and it will have to contract,
to pump some gravitational energy into thermal energy. The temperature of a contracting star
rises, and at some point heat released in nuclear reactions may balance the radiative heat losses.
If this happens then stellar contraction stops, and the star radiates away the energy generated in
nuclear burning. Whenever nuclear fuel is exhausted the ever present heat losses will force farther
contraction of the star. Therefore, it is the stellar radius that depends on the presence or absence
of nuclear burning, while the luminosity is controlled by the opacity."
(Yes, that takes a little longer to say, and all the supporting equations are given. In general, sites that
derive the relation get the reason right, those that rely on "truthiness" get it the opposite of right.)
Penn State: https://www.e-education.psu.edu/astro801/content/l7_p3.html
"Since higher mass means a larger gravitational force, higher mass must also mean that higher pressure is required to maintain equilibrium. If you increase the pressure inside a star, the temperature will also increase. So, the cores of massive stars have significantly higher temperatures than the cores of Sun-like stars. At higher temperatures, the nuclear fusion reactions generate energy much faster, so the hotter the core, the more luminous the star."
Open Stax: https://openstax.org/books/astronomy-2e/pages/18-4-the-h-r-diagram
"The most massive stars have the most gravity and can thus compress their centers to the greatest degree. This means they are the hottest inside and the best at generating energy from nuclear reactions deep within. As a result, they shine with the greatest luminosity and have the hottest surface temperatures."
Swinburne University: https://astronomy.swin.edu.au/cosmos/M/Main+Sequence+Lifetime
"Massive stars need higher central temperatures and pressures to support themselves against gravitational collapse, and for this reason, fusion reactions in these stars proceed at a faster rate than in lower mass stars. The result is that massive stars use up their core hydrogen fuel rapidly..."
Astronomy Notes: https://www.astronomynotes.com/starsun/s8.htm
"Massive stars have greater gravitational compression in their cores because of the larger weight of the overlying layers than that found in low-mass stars. The massive stars need greater thermal and radiation pressure pushing outward to balance the greater gravitational compression. The greater thermal pressure is provided by the higher temperatures in the massive star's core than those found in low-mass stars. Massive stars need higher core temperatures to be stable!"
So there you have it, complete consensus, totally wrong. Yet what student could possibly doubt the veracity of such an authoritative consensus? One who understands the core premise of science: be skeptical, think for yourself. If you want the correct reason, look at Wikipedia, or this Princeton course website:
Princeton University: https://www.astro.princeton.edu/~gk/A403/massive.pdf
"It is remarkable that we obtained the mass luminosity relation without any reference to the stellar
energy sources. This can be understood in the following terms. Within our approximation the
opacity of matter is constant (per unit mass), the photons diffuse out at the rate they can, which is
independent on stellar temperature or density. So the heat losses are fixed by the constant opacity. If
there are no nuclear energy sources then the star will be losing energy, and it will have to contract,
to pump some gravitational energy into thermal energy. The temperature of a contracting star
rises, and at some point heat released in nuclear reactions may balance the radiative heat losses.
If this happens then stellar contraction stops, and the star radiates away the energy generated in
nuclear burning. Whenever nuclear fuel is exhausted the ever present heat losses will force farther
contraction of the star. Therefore, it is the stellar radius that depends on the presence or absence
of nuclear burning, while the luminosity is controlled by the opacity."
(Yes, that takes a little longer to say, and all the supporting equations are given. In general, sites that
derive the relation get the reason right, those that rely on "truthiness" get it the opposite of right.)