astrophysics - When is it a good aproximation to consider a star to be an ideal gas?

I am currently taking a first course on stellar astrophysics, and I noticed that in some cases we use the ideal gas equation of state for stars, so we also use $gamma =5/3$. Of course it can only be applied where there is no nuclear reaction, so there is a limit in the temperature.

Also, if the radiation pressure if relevant, you have to consider the parameter $beta$ to calculate the total pressure and the adiabatic coefficient $gamma$. It is also incorrect (I think) if there are degeneration or relativistic considerations.

I don't know if I have to consider other factors before I can use this equation of state and value for $gamma$, maybe it is important if it is a convective or radiative area, or other factors I didn't consider.

My question is: can anybody tell me the limits of the ideal gas approximation? (quantitative better than qualitative but any help will be well received)

the sun - What would the effects be on Earth if Jupiter was turned into a star?

Before I start, I'll admit that I've criticized the question based on its improbability; however, I've been persuaded otherwise. I'm going to try to do the calculations based on completely different formulas than I think have been used; I hope you'll stay with me as I work it out.

Let's imagine that Lucifer becomes a main-sequence star - in fact, let's call it a low-mass red dwarf. Main-sequence stars follow the mass-luminosity relation:

$$frac{L}{L(s)} = left(frac{M}{M(s)}right)^a$$

Where $L$ and $M$ are the star's luminosity and mass, and $L(s)$ and $M(s)$ and the luminosity and mass of the Sun. For stars with $M < 0.43M(s)$, $a$ takes the value of 2.3. Now we can plug in Jupiter's mass ($1.8986 times 10 ^{27}$ kg) into the formula, as well as the Sun's mass ($1.98855 times 10 ^ {30}$ kg) and luminosity ($3.846 times 10 ^ {26}$ watts), and we get

$$frac{L}{3.846 times 10 ^ {26}} = left(frac{1.8986 times 10 ^ {27}}{1.98855 times 10 ^ {30}}right)^{2.3}$$

This becomes $$L = left(frac{1.8986 times 10 ^ {27}}{1.98855 times 10 ^ {30}}right)^{2.3} times 3.846 times 10 ^ {26}$$

which then becomes

$$L = 4.35 times 10 ^ {19}$$ watts.

Now we can work out the apparent brightness of Lucifer, as seen from Earth. For that, we need the formula

$$m = m(s) - 2.5 log left(frac {L}{L(s)}left(frac {d(s)}{d}right) ^ 2right)$$

where $m$ is the apparent magnitude of the star, $m(s)$ is the apparent magnitude of the Sun, $d(s)$ is the distance to the Sun, and $d$ is the distance to the star. Now, $m = -26.73$ and $d(s)$ is 1 (in astronomical units). $d$ varies. Jupiter is about 5.2 AU from the Sun, so at its closest distance to Earth, it would be ~4.2 AU away. We plug these numbers into the formula, and find

$$m = -6.25$$

which is a lot less brighter than the Sun. Now, when Jupiter is farthest away from the Sun, it is ~6.2 AU away. We plug that into the formula, and find

$$m = -5.40$$

which is dimmer still - although, of course, Jupiter would be completely blocked by the Sun. Still, for finding the apparent magnitude of Jupiter at some distance from Earth, we can change the above formula to

$$m = -26.73 - 2.5 log left(frac {4.35 times 10 ^ {19}}{3.846 times 10 6 {26}}left(frac {1}{d}right) ^ 2right)$$

By comparison, the Moon can have an average apparent magnitude of -12.74 at full moon - much brighter than Lucifer. The apparent magnitude of both bodies can, of course, change - Jupiter by transits of its moon, for example - but these are the optimal values.

While the above calculations really don't answer most parts of your question, I hope it helps a bit. And please, correct me if I made a mistake somewhere. LaTeX is by no means my native language, and I could have gotten something wrong.

I hope this helps.

Edit

The combined brightness of Lucifer and the Sun would depend on the angle of the Sun's rays and Lucifer's rays. Remember how we have different seasons because of the tilt of the Earth's axis? Well, the added heat would have to do with the tilt of Earth's and Lucifer's axes relative to one another. I can't give you a numerical result, but I can add that I hope it wouldn't be too much hotter than it is now, as I'm writing this!

Second Edit

Like I said in a comment somewhere on this page, the mass-luminosity relation really only works for main-sequence stars. If Lucifer was not on the main sequence. . . Well, then none of my calculations would be right.

genetics - Which patterns do I have to avoid when modifying the 3'-UTR?

If your goal is to avoid regulatory elements, I would not believe any prediction program, since new regulatory elements are found every day. The best way would be empirical and just clone and infect with a mutated 3'UTR and see if your gene's regulation is perturbed. You could at least swap the entire 3'UTR with that from another gene to see if there are even any gene regulatory elements in the 3'UTR.

general relativity - Gravitational field released during supernovae

What you are missing is that the shell theorem, which says that for a spherically symmetric mass distribution (we can count the pre- and post-supernova states as this for the sake of argument), that the gravitational field at some distance $d$ from the mass is the same as if all the mass were concentrated at the centre of the distribution.

Therefore the gravitational field due to the pre-supernova star is unchanged after the supernova at a distance $d$ until some of the mass (or equivalently in General Relativity, energy) as travelled beyond a distance $d$.

After a core-collapse supernova, if the pre-supernova star was say $15 M_{odot}$, then what might happen is that $1.4 M_{odot}$ is left as a neutron star remnant, whilst $10^{46}$ J of energy is released - mostly in the form of neutrinos travelling at almost the speed of light and the envelope of the star expanding outwards at tens of thousands of km/s and a kinetic energy of about $10^{44}$ J.

The neutrinos carry an equivalent rest mass of a mere $0.05M_{odot}$, so if you were on a (indestructable) planet in orbit around the supernova, then yes, after the main neutrino pulse has passed there would be a small decrease in the gravitational force felt towards the centre of the planet's orbit (not instantaneously, the neutrino pulse lasts some tens of seconds) that would result in an outward acceleration and the orbit widening slightly.

Sometime later (the orbital radius divided by the supernova ejecta speed) the main mass loss from the supernova would pass by and this would result in a drastic decrease in the gravitational force and a drastic widening of the orbit.

speed - Is Darkness Faster Than Light?

I've been looking around, and can't find a solid answer. Many websites say darkness is the speed of light due to the fact it is the absence of light, but other sources say otherwise, that in some cases darkness is actually faster in some cases. Could someone clarify this for me?

zoology - How do baby animals that primarily subsist on cellulose get their initial gut flora?

In the case of mammals like giraffes and koalas, is that bacteria common on the plants they eat so when a baby starts to try to stick something besides its mother's milk in its mouth, it can't digest the cellulose at all the first time, but along with the cellulose, into its mouth went some cellulose-digesting flora that then begins a lifelong colony in the animal's gut? Is that about how it happens?

the sun - Can our Sun become a black hole

No, the sun won't ever become a black hole.

The choice between the three fates of stars (white dwarf, neutron star, black hole) is entirely determined by the star's mass.

A star on the main sequence (like most stars, including our sun) is constantly in a balance between the inward pressure of gravity and the outward pressure of the energy generated by the hydrogen fusion that makes it "burn".¹ This balance stays relatively stable until the star runs out of whatever its current fuel is - at that point, it stops burning, which means there's no longer outward pressure, which means it starts collapsing. Depending on how much mass there is, it might get hot enough as it collapses to start fusing helium together. (If it's really massive, it might continue on to burn carbon, neon, oxygen, silicon, and finally iron, which can't be usefully fused.)

Regardless of what its final fuel is, eventually the star will reach a point where the collapse from gravity is insufficient to start burning the next fuel in line. This is when the star "dies".

White dwarfs

If the star's remains² mass less than 1.44 solar masses (the Chandrasekhar limit³), eventually gravity will collapse the star to the point where each atom is pushed right up against the next. They can't collapse further, because the electrons can't overlap. While white dwarfs do shed light, they do so because they are extremely hot and slowly cooling off, not because they're generating new energy. Theoretically, a white dwarf will eventually dim until it becomes a black dwarf, although the universe isn't old enough for this to have happened yet.

Neutron stars

If the collapsing star is above the Chandraskhar limit, gravity is so strong that it can overcome the "electrons can't overlap" restriction. At that point, all the electrons in the star will be pushed into combining with protons to form neutrons. Eventually, the entire star will composed primarily of neutrons pushed right up next to each other. The neutrons can't be pushed into occupying the same space, so the star eventually settles into being a single ball of pure neutrons.

Black holes

Black holes are the step beyond neutron stars, although they're worth discussing in a bit more detail. Everything, in theory, has a Schwarzschild radius. That's the radius where a ball of that mass would be so dense that light can't escape. For example, the Schwarzschild radius for Earth is about 9mm. However, for all masses smaller than somewhere between 2-3 times the mass of the sun, it's impossible to squeeze the matter small enough to get it inside that radius. Even a neutron star isn't massive enough.

But a star that becomes a black hole is. We don't actually know what happens to a star once it's become a black hole - the edges of the "hole" itself is simply the Schwarzschild radius - the point light can't escape. From outside, it doesn't matter whether the matter collapsed to the point that the neutrons started overlapping, whether it stopped just inside the radius, or whether it continued collapsing until it broke all known physical laws. The edges are still the same, because they're just a cutoff based on the escape velocity.

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¹ I'm ignoring the red giant phase here, since it's just a delay in the "run out of fuel" step. Basically, the core is helium "ash", while the hydrogen fusion process takes place further and further out. Once that runs out, you get a nova and the collapse continues.

² Likewise, I'm ignoring the mass that stars shed in their various nova phases. All given masses are based on the remnants left behind.

³Every source I've found for Chandrasekhar mass, except Wikipedia, gives 1.44 or 1.4 solar masses (which are compatible). Wikipedia gives 1.39, and gives at least one source to back that number.