Beware of "truthiness" in astronomy in seemingly reliable websites

[Ken G]

TL;DR Summary

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.)

 

[Drakkith]

(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!).

Can you elaborate? What temperature range do you mean when you say the core temps are 'similar'?

 

[DaveC426913]

Trust no one!
No one?
Except me!

 

[Ken G]

Can you elaborate? What temperature range do you mean when you say the core temps are 'similar'?

The need to be undergoing fusion means T is about 15 to 20 million K, give or take. So it's true that more massive stars have higher core T, but if you are just trying to understand if the gravity is stronger or weaker, the small T difference is not decisive. What is decisive is the huge difference in mass and radius.

To summarize the crux of the catastrophic error in all those websites and elementary textbooks, they assume high mass means strong gravity, forgetting that radius comes into play as well. The fact that all these stars are undergoing fusion means they have similar particle speeds in their cores, which means they have similar escape speeds, which means they have a similar ratio of mass to radius. (More quantitative analyses find that the mass to radius ratio rises slightly with mass, but again, this is not the decisive feature.)

Here's the point. If the mass to radius ratio is fairly constant, then Newton tells us the gravitational acceleration will scale inversely to the radius, which in this case means inversely to the mass. That's what we need to understand about main sequence stars, their characteristic gravitational acceleration is roughly inversely proportional to their mass, so high mass stars are weak gravity stars. Anyone teaching anything about stars should already know that, but they have settled for "truthiness" instead. Hence my warning, it's an easy mistake to fall into!

 

[Drakkith]

I assume high mass stars have higher pressures in their core, but is this actually true?
If so, is the higher pressure generated from more mass being held up and not greater gravity?

 

[DaveC426913]

Trust no one!
No one?
Except me!

I intended to be a little more nuance to this quip.

In one camp, there's Penn State, Open Stax, Swinburne U and Astronomy Notes.
In the other, there's Princeton and Ken G.
So it's 4 to 2, right? :)

If we can't trust authorities, then how can there be any authorities?

 

[Ken G]

I assume high mass stars have higher pressures in their core, but is this actually true?

No, high mass main sequence stars are generally low pressure objects, even in their cores. It relates to their weak gravity. (But yes, this is exactly the "truthiness" the thread is about, they seem like they should have high core pressure, and on that basis lots of educators are out there right now saying that they do! After awhile they are just quoting each other, in a classic example of the "echo chamber" effect.)

 

[Ken G]

I intended to be a little more nuance to this quip.

In one camp, there's Penn State, Open Stax, Swinburne U and Astronomy Notes.
In the other, there's Princeton and Ken G.
So it's 4 to 2, right? :)

If we can't trust authorities, then how can there be any authorities?

Don't overlook the key point: it's not just a question of tallying authorities, look at what is different in the Princeton argument from the others. That argument is based on a mathematical derivation of the relationship, not a "truthy" word salad argument. But the question you pose is an important one, how do we trust scientific authority? We remember Feynman's definition of science: http://www.feynman.com/science/what-is-science/
"Science is the belief in the ignorance of experts."

Now of course Feynman is not saying "ignore experts," merely that experts can be wrong, so science always checks and rechecks, and above all, asks us to think for ourselves. So look through the Princeton derivation (even the Wiki gives a similar derivation, https://en.wikipedia.org/wiki/Mass–luminosity_relation, the key point being that the relation comes from radiative diffusion and has nothing to do with fusion at all), see if it makes sense to you, and take from it its key insights. Then you can understand for yourself the mistakes made in all those other websites, and you don't have to take anyone's word for it.

 

[Vanadium 50]

we hear that higher mass stars have higher gravity

From whom?

If we are talking surface gravity (and what else could we be talking about) it goes in the other direction - the surface gravity of Proxima Centauri is 5x the sun's. On the other extreme, B Class main sequence stars will have a surface gravity about 1/3 that of the sun.

 

[Drakkith]

No, high mass main sequence stars are generally low pressure objects, even in their cores. It relates to their weak gravity. (But yes, this is exactly the "truthiness" the thread is about, they seem like they should have high core pressure, and on that basis lots of educators are out there right now saying that they do! After awhile they are just quoting each other, in a classic example of the "echo chamber" effect.)

Do you have any references showing stellar core temp, density, pressure, gravity, etc vs mass for main sequence stars? A quick google search didn't turn up much useful info for me.

 

[Ken G]

From whom?

All of the first few "authoritative" google hits that I quoted above, the ones intended for an elementary audience that does not expect to see derivations, and apparently, only expects something that sounds right but is completely wrong.

If we are talking surface gravity (and what else could we be talking about) it goes in the other direction - the surface gravity of Proxima Centauri is 5x the sun's. On the other extreme, B Class main sequence stars will have a surface gravity about 1/3 that of the sun.

Yes we agree, that's what I said above. Now look at what it says in the websites (and intro textbooks, by the way) that I quoted above. (These errors have been pointed out for decades, no one seems to bother to fix it. Truthiness at work.)

 

[Ken G]

Do you have any references showing stellar core temp, density, pressure, gravity, etc vs mass for main sequence stars? A quick google search didn't turn up much useful info for me.

You can run your own versions, at https://web.archive.org/web/20061006191021/http://shayol.bartol.udel.edu/~rhdt/ezweb/ .

Or if you like analytical approximations, assume that once you have the structure of one main sequence star, you can approximate any other using homology relations that ignore internal details like varying amounts of convection. If you do that, the virial theorem shows you that characteristic core thermal speeds are of order the surface escape speed. Since characteristic thermal speeds are always about the same in order for the protons to undergo fusion, that means characteristic escape speeds are also the same, which means the radius is roughly proportional to the mass, and the rest follows directly.

Or you can google the mass~radius relationship in main sequence stars (https://pages.uoregon.edu/imamura/321/122/lecture-5/radius.html) to get a more accurate version of this approximate proportionality. The level of detail you want varies between the approaches, but they all show the same basic fact: high mass stars have low core pressure.

 

[vanhees71]

TL;DR Summary: 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.)

Well, I have a pretty simple (and thus also pretty superficial) zeroth-order judgement about "scientific texts": "Formula density", i.e., the more math there is in an explanation the more you may expect to get some sound and solid explanation. As I said, that's of course very superficial: You have to first read carefully, whether the math really seems to make sense, and if you use any result in your own work, you should be sure to check the math in detail.

Exception to the zeroth-order judgement rule: If a text is written in M$ Word, even with a high formula density, it's likely not to be very solid ;-)).

 

[Ken G]

The problem, though, is that some audiences use the opposite criterion: their probability of reading and being influenced by a scientific text is inversely proportional (to a high power) to the formula density! As such, there is a need for the "translators," the educators who take the formal texts and "distill" them into words that seem reasonable. Then you get into the issue of, if you are not being quantitative, how accurate does the distillation need to be? That is a subjective issue, but we can agree that being the opposite of correct is not a good distillation! Truthiness can lead to distillations that are the opposite of correct, like the ones I quoted above.

The trick is to capture the essence without the equations, all while avoiding the "echo chamber" pitfall. I applaud those who make the effort, but what I object to is those who don't seem to care if what they say makes any real sense at all. They, like Calvin's dad, seem to be saying "you'll never understand the true answer, so I'll just feed you something you will accept, and it doesn't matter as long as you think you understand it."

 

[vanhees71]

"Elementarizing" is an art of its own, and expressing physics in non-mathematical language is particularly difficult. Sometimes you even have the phenomenon that there are such "explanations" in the scientific community, which are repeated for generation after generation of physicists, and nobody dares to ask about them, although also nobody really understands them. As in any human endeavor also in science there's a huge psychological component in it. The right attitude for any natural scientist is skepticism!

 

[Ken G]

And the complement to skepticism is the desire to "elementarize" our own understanding, a process that is essential if science is to be anything more than simply training machines to make correct predictions. Here is how I would elementarize the mass-luminosity relation, and I hope that someday words to this effect is what we find in the elementary sources:

Most main sequence stars are basically big leaky buckets of light. The "bucket" is largely hydrogen gas, and it is very bright inside because a history of gravitational contraction has made the gas extremely hot. The light is trapped inside the bucket by the fact that it cannot travel very far before being absorbed or scattered, but it does gradually leak out, on timescales like hundreds of thousands of years. The greater the volume of the star, the bigger the bucket, and the more light it contains. The luminosity depends on the amount of light in the bucket, divided by the time it takes to leak out. Since the time is roughly similar for all main-sequence stars, the luminosity is approximately proportional to the volume of the star. Higher mass main-sequence stars have a larger radius, so a much higher volume, so contain much more light in the bucket. Hence that light leaks out at a much faster overall rate, and the higher mass star has a much higher luminosity.

That's every bit as simple as the wrong idea that higher mass stars have higher core pressure, and above all, it never mentions fusion. This is important, because correct derivations of the mass-luminosity relation do not require any fusion physics, the relation has really nothing to do with fusion. It is actually quite obvious that fusion cannot matter, because stars with masses like the Sun, or higher, set their main-sequence luminosity long before they actually reach the main sequence. That means the relation holds long before the star even knows fusion exists. (What's more, an early derivation of the mass-luminosity relation was obtained by Eddington, and guess how much he knew about fusion at the time! Given that, isn't it amazing that so many elementary textbooks say that fusion is what controls the mass-luminosity relation?)

There is a subtlety for the careful reader who wants to dig deeper. The idea that the radius of a star is larger if the mass is larger is a place where fusion physics implicitly appears, because the radius is roughly proportional to the mass if the core is at fusion temperature. However, the mass-luminosity relation does not require that, because by coincidence it holds for any radiatively diffusive star, regardless of its radius. This brings us into another whole issue, which is whether we regard the mass-luminosity relation as an empirical relation involving only main-sequence stars (in which case the above answer holds, though there is some implicit fusion physics there), or if we regard it as a theoretical connection between all radiatively diffusive stars (which does not require any fusion, but leaves out main-sequence red dwarfs, even though they approximately maintain the relation, again by coincidence!). Good luck navigating whether the mass-luminosity relation should be a main-sequence law, or a law obeyed by radiatively diffusive stars!

 

[sophiecentaur]

If so, is the higher pressure generated from more mass being held up and not greater gravity?

Is this related to the fact that the more massive stars have much shorter development time scales?

From whom?

If we are talking surface gravity (and what else could we be talking about) it goes in the other direction - the surface gravity of Proxima Centauri is 5x the sun's. On the other extreme, B Class main sequence stars will have a surface gravity about 1/3 that of the sun.

Isn't the relevant quantity Gravitational Energy? Would surface gravity be directly affecting the internal pressure near the core?

 

[Ken G]

Is this related to the fact that the more massive stars have much shorter development time scales?

The way that works is, because light diffuses out so rapidly, fusion dials itself up to match what is being lost, and that in turn causes the lifetimes to be short.

Isn't the relevant quantity Gravitational Energy? Would surface gravity be directly affecting the internal pressure near the core?

There are certainly several important gravity related quantities to track, like escape speed, gravitational energy, and what gets called "gravitational pressure." The latter has units of pressure, and is characterized by the "weight of the star" divided by the "cross sectional area" of the star. So it's that latter quantity we've been talking about, where the weight of the star is characterized by the square of the mass of the whole star, divided by the square of its radius, and the cross section scales with an additional square of the radius. Thus if you compare one main sequence star to another, the pressures everywhere in both will be similar, just scaled up by the ratio of the masses squared, divided by the ratio of the radii to the fourth power. That's the factor that is much smaller in high mass stars, which explains why their core pressures are similarly smaller. (That's the kind of reasoning embedded in that derivation from the Princeton website.)

To have a physical picture of what the "weight of a star" is, it would not make sense to put the star on a scale, as that requires some external gravity. Instead, imagine the star is sliced in half like a watermelon, and place one half on top of the other, letting their self gravity squeeze the halves together. That squeezing force characterizes the "weight of the star", and dividing it by the cross sectional area then characterizes the "gravitational pressure" that the gas pressure must balance. We don't worry about factors like pi and 2 here, because we are comparing two stars of different mass, and assuming all those factors work out the same for both stars, so cancel out in the comparison.

 

[Vanadium 50]

Isn't the relevant quantity Gravitational Energy?

Could be. Relevant for what? Is the relevant quantity for a car its top speed, its MPG, the number of passengers it can take or its color? Yes.

The gravity at the surface as well-defined. The gravity at the center is well-defined too, but unfortunately is zero. In between you can look at a g vs. r curve, but you can only compare two curves if one is uniformly above the other. I haven't looked at such plots (what would be the point?) but suspect this does not happen - a handwavy argument suggests that they always cross somewhere.

 

[Ken G]

The standard simplification is to imagine a "homology," meaning that the curves you mention have the same shape, simply scaled up by the amounts I've been talking about.

 

[ohwilleke]

Trust no one!
No one?
Except me!

Random strangers on the Internet are widely acknowledged to be the most accurate source of information on just about everything.

 

[Ken G]

Isn't that more or less the "Wikipedia" concept? (Not to mention the forum concept as well....) The interesting point is, we already know to be skeptical of "random strangers," the purpose of the above quoted websites is to show that we also have to be skeptical of university professors! (Albeit not nearly as often, of course.)