Cham

@ProfRob, I'm having some issues with the equation of state for the polytropic fluid, in a general case. There are some inconsistencies, especially in the relativistic and ultra-relativistic regimes. I'll have to ask a new question about this.

Forget what I wrote...

Hmm, correction: for a sufficiently low central pressure, the radius isn't infinite.

@ProfRob, I remade my numerical integration of the TOV equations for the ultra-relativisitc polytropic fluid (adiabatic index $\gamma = \frac{4}{3}$), and I'm finding something odd: the star appears to have an infinite radius, whatever the central pressure. This is odd with the non-relativistic Newtonian equation which gives an infinite radius only for $\gamma \le \frac{6}{5}$. Do you think this is true?

@ProfRob, the pressure at 0.5 is just one example. I can change that value from 0 to 10 (or even more), in the units chosen (for which $G \equiv 1$ and $\kappa \equiv 1$. The shape of the pressure curve is "typical".

@ProfRob, how do we know that the infinite radius for $𝑛 > 5$ is also valid for the relativistic regime? The previous argument is from Newtonian's gravity only.

About a theoretical proof of infinite radius for $𝑛>5$, I found the following in my notes: For a newtonian polytropic fluid, we can get the potential energy:\begin{equation} 𝑈=−\, \frac{3}{5−𝑛} \, \frac{𝐺𝑀^2}{𝑅}.\end{equation} So this expression implies $𝑛<5$ to get $𝑈<0$ (gravity is attractive). It gives $𝑈>0$ for all $𝑛>5$, unless $R\rightarrow\infty$. So indeed $\gamma=1+\tfrac{1}{𝑛} = \tfrac{6}{5}=1.20$ is "special". I'm not sure this is also valid for the relativistic (non-newtonian) regime but it makes sense.

@ProfRob, I updated my question with two plots for $\gamma = \frac{4}{3}$ (which is larger than your $\gamma = \frac{6}{5} = 1.20$ ($n = 5$). From this plot, I don't see how I could find a finite radius.

@Quillo, $\gamma = 5/3$ is certainly larger than 1! I'm not really interested in $\gamma < 1$ and $\gamma > 2$ which are physically very weird (my code shows all sorts of weird singularities with these values). I don't think the cutoff is related in any way to the iron layer. On the contrary, the cutoff is defined at very low pressure and density, which isn't in the core of the star.

@Quillo, I'm using all $\gamma$, from $\gamma \approx 1$ up to $\gamma = 2$ (even if non-realistic, to show the effects). Since I'm not using any value of $\kappa$, the output does not depend on the matter type or the micro-physics (except from the $\gamma$). I'll find the radius value in km only if I switch back from unitless variables ($\tilde{p}$ and $\tilde{r}$) to unitfull variables ($p$ and $r$), according to the description below equation (4) (please, read again!).

@Domen, yep! This last command appears to solve my issue! Thanks a lot! Case appears to be solved now. Again, thanks for all the help!

I just noticed that only 4 spheres cause the warning messages. There's no warning with 3 spheres or less.

I can give you the whole code if you wish (I think it's a cool one!). It draws colorfull curbes and balls in 3D, and the manupilation is very nice...

6 spheres, sorry, not 8

Yep, the output is good, but I get several error (warning) messages about precision.

I'm just watching if the output is normal...

I'll give you the modification to the code (it's simple)

Compiling right now with 8 magnetized spheres. I'm getting the error messages

Wait a minute...

But if I add more sphere, lets see

I tried with three spheres, no messages

I do use this command on all of my codes. Still getting the error messages. Maybe not related to the conditions, I don't know.

I want to solve these error messages. I don't know how.

If I use {cond = (r[s, k] > 25 || Apply[Or, Table[r[s, kk] < 1, {kk, 1, Nspheres}]])} inside the With environment, I get exactly the same output (which is nice) as using the StoppingTest line, and still get the error messages: "Event location failed to converge to the requested accuracy or \ precision within 100 iterations between ..."

@Domen, then what about the r[s, k] < 1 condition? Why not put it alongside with >25? I don't understand this part

Why is the cond event declared inside a With environment, and not the other conditions?

Sorry, I have to go now. Thanks for the help. But I'm still not convinced that I have the proper solution.

I understand, but at least the normal display isn't affected by the text, while it is for the inline equations.

In other word: I don't want $y_2(x) = a x^2$ to be affected by the text flowing around it.

I mean the same indices and exponents as in a normal display, and the same spacings...

Like I said, I don't like the variable output of $y_2(x) = a x^2$, which changes depending of the text around it. I want that equation $y_2(x) = a x^2$ to have the same output whatever the text around it.

Two ways?

? You mean \everymath{\displaystyle}, plus your solution? There is no other way?

But I also need to use $\displaystyle{y_2(x) = a x^2}$ to get proper vertical position of all indices and exponents. I simply need to do all these at the preamble level.

Geez this is complicated! For example, I write (inside text) $y_2(x) = a x^2$. The output is frequently variable, depending of the text around it. The spacings may be too strong (dilated, or compressed), because of the text itself. I don't want the text to have an effect here, so I could use ${y_2(x) = a x^2}$, which fixes the spacing.

If so, then I would have to use braces (i.e ${...}$) for all my inline math instead?

Ahaa ! This is important : does your solution also change the display of any other maths, under the text?

Should I put your commands inside the \everymath{\displaystyle} ?

Your solution may be the right one I'm looking for. But I was wondering if it was the only one. Especially since I also need to use the \everymath{\displaystyle} command.

Yes.

I do't understand. Sorry, there may be a communication issue. Currently, I could get the right output if I simply use $\displaystyle{...code in there...}$ for all inline maths. I want to achieve the same by a global preamble command.

Again, I just want a global preamble command (or set of commands) to do the same output as using $\displaystyle{...}$ in all local inline equation.

Ok. But my question is then the same as my previous comment.

Actually, this is my issue: using \everymath{\displaystyle} in the preamble doesn't give the exact same output as $\displaystyle{...}$ for all inline math expressions.

By "fixing", I mean to "keep constant". Or if you prefer, to display as in a full math display (below text). Using local ${...}$ fixes my issue, or more specifically $\displaystyle{...}$. I just want to do the same in a global way, in the preamble (in case I change my idea), not in a local way. So what should be a proper preamble command to do the same effect as $\displaystyle{...}$ ? Using only \everymath{\displaystyle} doesn't give the same effect (the spacings aren't the same).

For any inline expression, I only type small things (which may have indices and exponents, and small fractions, small relations). All bigger equations are displayed below the text. When I use the normal way, the math horizontal spacings are modified by the text. I need to prevent that to happen.

I don't understand greg's answer yet. All I need, is the fix my small inline math expressions as they would appear in a full display (between paragraphs). If I use the normal way with variable inline spacings, and variable position of indices and exponents, the math expressions are frequently very ugly.

@DavidCarlisle, ok. But I don't want the text to have any effect on the inline math spacings. I need them to stay the same as if they were displayed between the text paragraphs. The text spacings should be variable in the normal way, but the inline math spacings should stay the same as if they were displayed under the text.

@DavidCarlisle, I use displaystyle for inline equations because I want all the indices and exponents to stay the same as in the bigger equations shown between text paragraphs. I also want the horizontal spacing to be the same. I'm only typing small math expressions inline. Never "big" expressions.