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The Sun Had Sisters
[TheBORG] writes to mention a Space.com article about the Sun's departed solar siblings. Our own medium-sized yellow star was far from alone when it was formed, with hundreds of fellow solar bodies and a supernova to keep it company. From the article: "The evidence for the solar sisters was found in daughters--such as decayed particles from radioactive isotopes of iron--trapped in meteorites, which can be studied as fossil remnants of the early solar system. These daughter species allowed Looney and his colleagues to discern that a supernova with the mass of about 20 suns exploded relatively near the early Sun when it formed 4.6 billion years ago; and where there are supernovas or any massive star, you also see hundreds to thousands of sun-like stars, he said. The cluster of thousands of stars dispersed billions of years ago due to a lack of gravitational pull, Looney said, leaving the sisters 'lost in space' and our Sun looking like an only child ever since, he said."
No, as I understand the theory (and IANAA - well except an occassional amateur one) if a supernova explodes in or near a gas cloud the shockwave initiates star formation.
No. The shockwave from the supernova produced localized density increases in a nearby or surrounding gas cloud. These density increases pushed the local gravitational field over the level at which the gas begins to accrete into what will eventually become a star. Such shock waves are the main cause of starts being formed, and the reason why there are "star nurseries" - volumes of space in which a large number of new stars are being born.
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Dadoo is correct. A very massive star has to have a hotter core at its center in order to support its heavier stellar mass (the hotter the gas, the higher the gas pressure, and hence the more effective to support its own weight in order not to collapse into a singularity, i.e., a blackhole). And the rate of nuclear reaction is often proportional to a higher power of Temperature at the core. That means the hotter the core is, the faster it is to synthesize heavier elements from proton to Helium.
As the same star evolves, it depletes hydrogen (proton) soon at the core. But because the star is still massive, it enables to burn helium, then carbon, oxygen, nitrogen, and eventually it starts burning more heavier elements via nuclear processing (til iron -- Fe -- which cannot be burned to generate nuclear energy).
This heavier element synthesis is accelerated by high temperature and pressure (basically) at the core of a star. For a very massive star (Mass ~ 100 sun) it lives only about a few million years before it begins to show the sign of aging (heavier metallic elements in its atmosphere). And when these stars die, their explosions would disperse these heavier elements throughout its neighboring space (also upon explosion, an ample flux of neutrons would bombard other atoms and eventually the atoms trap the neutrons to form heavier elements than Fe; Strontium, uranium, plutonium and gold are good examples of such process).
In a small star like the Sun, the synthesis process takes place very slowly (in the time scale of a few billion years). So it's only natural that astrophysicits think today that there must have been a lot of very massive stars formed in the early days of the Universe to explain its metallicity level seen today.
Star clusters fall apart from lack of sufficient gravitational attraction all the time. This shouldn't be surprising. Just because some stars are rotating around each other for a while doesn't mean the orbits are stable.
The article doesn't say exactly, but there's some easy inferences. We were part of a star cluster. There was a large star in the cluster, providing a large amount of gravitational attraction. That star then went nova, shedding a large portion of its mass. Ta-da, there is no longer enough gravity to hold the cluster together, and it flies apart.
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You're both wrong, of course. The order of discretization has nothinig to do with this, the naive choice of coordinates does. It is easy to do a fine (first-order!) simulation if you choose appropriate coordinates: In the case of a circular orbit, for example, the phase-angle of the orbit is all you need - it is the only thing the Lagrangian depends on explicitly and only linearly. Pick phi and r as your coordinates and the very first line on your page will be "d/dt r=0".
In a more general case, energy or angular momentum are usually good coordinates to use, because the Lagrangian does not depend on them. And thus they are conserved simply by inspection of the equations. And thus the only way you could ever lose them is by making a programming mistake.
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You're both wrong, of course. The order of discretization has nothinig to do with this, the naive choice of coordinates does.
This "naive" method is PRECISELY what we are discussing. Look at my comment. Notice my usage of the word "naive." Notice that you are not following the topic.
This thread is not about the impossibility of an energy-conserving first order method. It is SPECIFICALLY about the naive cartesian Euler method, which is what I presumed the OP implemented.
But hey, you get to show off your mad skillz... Fine.