How many stars died to make me?

[Diadem]

As most people here will know, all the heavy elements that make up the earth, and our bodies, were formed during supernova explosions. The earliest, heavy, stars were almost pure hydrogen, and when they died, they formed heavy elements and scattered them across the galaxy.

So today I suddenly had a question: How many of those stars did it take to make the earth? Do the elements that make up the earth come from just a singular star, or many? Is it possible to get an order-of-magnitude estimate?

"I was forged in the heart of a billion dying suns" would be a rather cool line. I wonder if it's true though. Any astrophysicists around who can shine their light on this?

[Xanthir]

Many stars, definitely. The Sun is a third-generation star - do you count only the stars that died immediately before the sun, or the ones that died to make *them*, too?

[starslayer]

Ooh, tough question. The trouble with answering this is that different elements come from different places. For example, Type II (core collapse) supernovae spew alpha elements (take C and keep adding alpha particles) everywhere, but virutally no iron. Type Ia's, since they leave no central remnant, make tons of iron peak elements. S-process elements come from AGB stars and supernovae, and we don't know where the hell the r-process elements come from.

Still, we can try to come up with an order of magnitude estimate. Since supernovae are the main seeds of the process, we should be able to ignore the other nucleosynthesis sites for the purposes of the calculation. Therefore, we need to calculate how many stars had to explode to bring the stellar neighborhood up to solar metallicity. To do that we need to know the star formation history of the galaxy, which is difficult at best. The present star formation rate (SFR) and supernova rate (SNR) are 1 solar mass/yr and about 2 per century at present. Both of these were higher in the past. The Milky Way was probably never a starburst galaxy, but we can't be sure. As a complete guess, we'll say the SFR was a constant 3 solar masses/yr between the galaxy's formation and the Sun's (so, ~7-8 billion years). This feels right to me, though this average rate was probably higher. Still, we can get a nice lower bound on how SNe it took to enrich the galaxy. SFR and SNR are approximately linearly correlated, so we would expect ~5 SNe per century over the lifetime of the Milky Way. Run for the required time, and we get about 350-400 million SNe in the Milky Way before the Sun showed up.

However, the galaxy is not well-mixed, so as you go further in towards the galactic center, the metallicity gets higher, as those parts of the galaxy are denser and older. Since metals from the bulge don't make it out to the suburbs where we are, we need to revise our estimate downward. I've no idea by how much, but a factor of 5-10 seems reasonable enough, though this is still a complete SWAG. This would mean you need somewhere in the neighborhood of 50 million SNe to give you the conditions to replicate the Earth we know and love.

[Diadem]

Many stars, definitely. The Sun is a third-generation star - do you count only the stars that died immediately before the sun, or the ones that died to make *them*, too?

Only the one that produced the elements found on earth today.

Every single atom of my body (except hydrogen and traces of other light elements) was formed in a supernova. I have a lot of atoms in my body, many were no doubt formed in the same explosion, but not necessarily all of them. So how many different supernovae were they formed in?

Some of those atoms may have been part of multiple stars, but they were only formed in one. I guess a very small fraction could have been formed in stages, with the first star fusion hydrogen to something below iron, and that element subsequently being fused into a heavy element by a later star. But the final atom will still have formed in 1 single star.

[Diadem]

Ooh, tough question. The trouble with answering this is that different elements come from different places. For example, Type II (core collapse) supernovae spew alpha elements (take C and keep adding alpha particles) everywhere, but virutally no iron. Type Ia's, since they leave no central remnant, make tons of iron peak elements. S-process elements come from AGB stars and supernovae, and we don't know where the hell the r-process elements come from.

Still, we can try to come up with an order of magnitude estimate. Since supernovae are the main seeds of the process, we should be able to ignore the other nucleosynthesis sites for the purposes of the calculation. Therefore, we need to calculate how many stars had to explode to bring the stellar neighborhood up to solar metallicity. To do that we need to know the star formation history of the galaxy, which is difficult at best. The present star formation rate (SFR) and supernova rate (SNR) are 1 solar mass/yr and about 2 per century at present. Both of these were higher in the past. The Milky Way was probably never a starburst galaxy, but we can't be sure. As a complete guess, we'll say the SFR was a constant 3 solar masses/yr between the galaxy's formation and the Sun's (so, ~7-8 billion years). This feels right to me, though this average rate was probably higher. Still, we can get a nice lower bound on how SNe it took to enrich the galaxy. SFR and SNR are approximately linearly correlated, so we would expect ~5 SNe per century over the lifetime of the Milky Way. Run for the required time, and we get about 350-400 million SNe in the Milky Way before the Sun showed up.

However, the galaxy is not well-mixed, so as you go further in towards the galactic center, the metallicity gets higher, as those parts of the galaxy are denser and older. Since metals from the bulge don't make it out to the suburbs where we are, we need to revise our estimate downward. I've no idea by how much, but a factor of 5-10 seems reasonable enough, though this is still a complete SWAG. This would mean you need somewhere in the neighborhood of 50 million SNe to give you the conditions to replicate the Earth we know and love.

Thanks. This is the kind of reasoning I am looking for. Now we just need to reduce the amount of guesswork in there

That is considerably more than I expected. And judging from your figures it sounds like a pretty low estimate. Only a factor 2.5 increase in supernova rate seems rather low. I always thought star formation was much, much higher in young galaxies? Also, why would heavy elements from the galactic centre not reach us? Their velocity is a significant fraction of the speed of light, well above the escape velocity. And there's plenty of time. Traversing a 100K lightyears at relativistic speed does not take long, on the time scales involved.

[mfb]

I think the higher metallicity just comes from the fact that objects closer to supernovae get more of its material.
To stop it, the galaxy would have to have massive gas clouds which can stop high-energetic particles. One radiation length for hadrons is something like 1m of liquid water or 1000kg/m^2. Give or take an orders of magnitude, this area density would require something like 50000 solar masses per light year squared to give a relevant shielding. The visible 2*10^11 solar masses of our galaxy could be arranged in a sphere with a radius of 600 light years with the shield density given above. However, this is certainly not the mass distribution in the milky way, and has never been. Therefore, I expect that most high-energetic particles of a supernova can reach all parts of the galaxy and even leave it.

If a supernova in the galactic center distributes 1 solar mass of iron in a sphere with our distance to the galactic center as radius, we get 10^13 particles per square meter. Even if 99,99999% (or whatever) do not reach the solar system, a lot of them stay here.

=> I would expect atoms from nearly every supernova in the milky way in our solar system. And probably some from other galaxies, too.

[Diadem]

=> I would expect atoms from nearly every supernova in the milky way in our solar system. And probably some from other galaxies, too.

Provided it's not too diluted. Let's calculate that. Let's calculate how far away a supernova can be before it no longer contributes sufficiently to our solar system.

Well for a start wikipedia says that the sun was formed from a part of a molecular cloud approximately 1 pc across. One pc^3 = 35 ly^3. So let's take that figure. Say a supernova happened D ly away, and has had sufficient time to spread out to us. Let's assume that a supernova spreads its heavy elements uniformly over the region of space its shock wave has reached (it travels at about 10% of the speed of light). This is of course a huge assumption, but if it's not true, that just means some regions gain more from one supernova and less from another, so I hope it'll still average out approximately right.

The sun is 99% of the mass of the solar system and about 2 * 10³⁰ kg. But I'm mostly heavy elements, the sun is mostly hydrogen and helium. Wikipedia tells me the sun is 0.77% oxygen (in mass). Assuming I'm 100% water I'm 89% oxygen (again in mass). If the average human is 80kg, then about 1 in 2 * 10²⁶ atoms of oxygen in the solar system is in me. So a supernova needs to leave 10^26 oxygen atoms in an area of 35 ly^3 for it to leave a trace in me.

If a supernova produces K solar masses of heavy elements, all as oxygen, then it produces it produces (K * M_sol * N_A / 16) atoms of oxygen. So for the total number of oxygen atoms per pc^3 we have:

(K * M_sol * N_A / 16) * 35 / (4/3 * Pi * D)³) = K/T^3 * 6 * 10⁵⁶.

I have no idea how many heavy atoms are created in a supernova. But let's be conservative and say K=1. We wanted 10^26 oxygen atoms in our solar system. That gives D = 10^10. About 10 billion lightyears. That's much more than I expected. Heavy elements are ejected from a supernova at about 10% of the speed of light, so the heavy elements from a supernova that far away would take 100Gy to reach us. Longer than the age of the universe.

So my calculation breaks down. Alas. But that does mean that pretty much every supernova in the universe that has had time to reach us will have left sufficient heavy elements in our solar system for some of them to be in my body. And the margin of error is pretty big, so that conclusion holds even if some of my estimates were off.

So we need to look at a HUGE portion of the universe. I can't say how much, the time and length scales involved are such that the effect of expansion of the universe becomes significant. Someone better versed in cosmology may do that calculation.

But with a few hundred million supernova in our galaxy alone, the total number will then number in the trillions. Wow.

[lorb]

Let's assume that a supernova spreads its heavy elements uniformly over the region of space its shock wave has reached (it travels at about 10% of the speed of light). This is of course a huge assumption, but if it's not true, that just means some regions gain more from one supernova and less from another, so I hope it'll still average out approximately right.

I think this assumption is too much of a stretch, especially since supernovae are known to be asymmetric, and there may be plenty of objects/forces around which do have an impact where the matter goes.

[starslayer]

That is considerably more than I expected. And judging from your figures it sounds like a pretty low estimate. Only a factor 2.5 increase in supernova rate seems rather low. I always thought star formation was much, much higher in young galaxies? Also, why would heavy elements from the galactic centre not reach us? Their velocity is a significant fraction of the speed of light, well above the escape velocity. And there's plenty of time. Traversing a 100K lightyears at relativistic speed does not take long, on the time scales involved.

Remember, I quoted a (completely made up) average SFR over the lifetime of the galaxy. It has for the most part been slowly, constantly decreasing over the lifetime of the galaxy, and was indeed much higher when the Milky Way was young. And SNR goes hand-in-hand linearly with SFR, assuming the initial mass function¹ (IMF) is essentially universal. All signs so far point to the idea that it is.

Now, starburst galaxies do exist that can have SFRs stretching up into the hundreds of solar masses/yr (this might not sound like it, but it is a truly prodigious SFR - it means there are at least 1-2 SNe every year in that galaxy!), but this is not sustainable. They very quickly exhaust most of their star formation capable gas (usually within a few tens of Myr), and essentially shut off. For that reason, I think it is fairly reasonable to assume an average SFR only a few times higher than the current Milky Way SFR.

As for supernova ejecta, while it is going much faster than galactic escape velocity, it runs into the ISM and slows down fairly quickly; we can see remnants due to the shock waves that they generate as they expand. IIRC, significant amounts of ejecta don't get more than a couple of kpc away from the initial explosion site. Also remember that most supernovae, especially core-collapse supernovae, occur in giant molecular clouds (GMCs) and their immediate remains, regions where the ISM is much denser than normal. The supernovae blow bubbles in the ISM that are typically a few hundred pc across, and the majority of the progenitor star's mass remains within the bubble. I suppose, however, I should not have categorically declared that absolutely no material from bulge star formation makes it out to the solar neighborhood and beyond, but it isn't a very high fraction of the material that does. For the purposes of the order of magnitude calculation, it can shuffled under the rug.

Now, as a separate question, if you just want to ask how many SNe it takes to just make the amount of heavy elements we find in the solar system, that's much smaller. To get the correct abundances, you need a combination of core collapse and Type Ia SNe, plus a few AGB stars to make the s-process elements. But if you just want to have enough material to make our solar system, without caring about getting the absolute right abundances, it's probably only one, since core collapse SNe spew out many solar masses worth of ejecta, quite a bit of which is heavy elements. If you want to hedge your bets, make it two to throw in a 1a for a solar mass or two of iron peak elements.

I think this assumption is too much of a stretch, especially since supernovae are known to be asymmetric, and there may be plenty of objects/forces around which do have an impact where the matter goes.

Nah, Diadem's got it. They're symmetric enough that it all averages out, especially when you have a lot of them going off in a relatively short time (~10 Myr or so). Most of the SNe in the galaxy, like 70-75%, occur in OB associations,

¹The IMF is basically the distribution of stellar masses. It describes quantitatively how many stars of a given mass are formed in a given volume. The first good estimate of this was done by Salpeter in the 50's. A Salpeter IMF has N(M)dM, the number of stars with mass between M and M+dM, proportional to M^2.35. More recent IMFs have a shallower slope at the low mass end below 1 solar mass; this better fits the observed stellar populations we see.