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I am working on a code for galactic dynamics simulation. Initial conditions (positions and velocities) are based on a model with a massive dark halo and a thin exponential disk.

Now, from the following paper "Tidal tails in CDM cosmologies", I am studying the response of the dark matter profile.

Here's a capture of the article part that confuses me :

Response of dark matter profile

Firstly, I don't understand how one gets equation (16) from the fact that angular momentum of individual dark matter orbits is conserved, i.e :

\begin{equation} r_{i}\,M(r_{i})=r_{f}\,M_{f}(r_{f})\quad\quad\text{(eq 16)} \end{equation}

" where $r_{i}$ and $r_{f}$ are the initial and final radii of some dark matter mass shell, $M(r)$ gives the initial NFW mass profile, and $M_{f}(r)$ is the final cumulative mass profile after the disk is formed."

Question 1) Is (equation 16) got by the momentum theorem ? ($\dfrac{\text{d}\vec{L}}{\text{d}t} = \vec{OM} \times \vec{F}$)

And how to prove it, i.e what's the demonstration to get (equation 16) ?

I don't understand what's the difference between $M(r)$ and $M_{f}(r)$.

After, it is said that $M_{f}(r)$ is the sum of the cumulative mass of the disk and the dark mass inside the initial radius and author writes (equation 17 on paper) :

\begin{equation} M_{f}\,(r_{f})=M_{d}\,(r_{f})+(1 - m_{d})\,M(r_{i})\quad\quad\text{(eq 17)} \end{equation}

I would like to connect this equation with the total mass of galaxy which could be written (according to me) as :

\begin{equation} M_{total}\,(r) = M_{d}\,(r) + M_{DM}\,(r) = m_{d}\, M_{DM}\,(r) + M_{DM}\,(r) = (1+m_d)\, M_{DM}\,(r) \end{equation}

with $m_{d}$ the ratio between disk mass and halo mass.

Question 2) Could you help me to connect the total mass with this equation (17) ?

After with equation (18), author of paper writes that the final profile $M_{h}(r)$ of the dark matter halo is then given by :

\begin{equation} M_{h}\,(r) = M_{f}\,(r) - M_{d}\,(r) \end{equation}

Question 3) So should I conclude that $M_{f}\,(r)$ is the total mass of galaxy (disk + halo) as a function of $r$ ?

Any help would be great,

Regards

UPDATE 1 :

I have contacted the author of paper "Tidal tails in CDM cosmologies", i.e Volker Springel and here are its comments :

Question 1*) Answer: For simplicity, you can consider that all particles are on circular orbits. The initial circular velocity of a particle is $$v_i = \sqrt{(G M(r_i)/r_i)}$$, the final one is $$v_f = \sqrt{(G M(r_f)/r_f)}$$. The corresponding angular momenta are $$L_i = m r_i v_i$$ and $$L_f = m r_f v_f$$. Now, the angular momentum is an adiabatic invariant when the halo is slowly compressed, hence $$L_i = L_f$$. The latter equation then gives eqn (16) when one plugs in the velocities.

Question 2*) "I don't understand what's the difference between $M(r)$ and $M_f(r)$ ?"

Answer:

$M_i(r)$ is the initial and $M_f(r)$ the final cumulative mass profile.

Question 3*)

I would like to connect this equation with the total mass of galaxy which could be written as :

$$ M_{total}(r) = M_d(r) + M_DM(r) = m_d M_DM(r) + M_DM(r) = (1+m_d) M_DM(r)$$

Answer : This can only be written for the initial distribution in this form, where the assumption is that the mass that will later make up the disk is distributed proportionally to the dark matter.

Question 4*) Could you help me to connect the total mass with this equation (17) ?

After with equation (18), you write that the final profile M_{h}(r)$ of the dark matter halo is then given by :

$$M_h (r) = M_f(r) - M_d(r)$$.

So should I conclude that M_f(r) is the total mass of galaxy (disk + halo) as a function of r ? < Answer : "Yes."

In conclusion, I can solve my question 1*) but not question 2*) nor question 3*)

Right now, my goal is to demonstrate how to get :

$$\begin{equation} M_{f}\,(r_{f})=M_{d}\,(r_{f})+(1 - m_{d})\,M(r_{i})\quad\quad\text{(eq 17)} \end{equation}$$

Any help to acheive this is welcome, thanks

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  • $\begingroup$ Yes, that is simply the angular momentum of the shell written down, and using the fact that it's conserved. $\endgroup$ – AtmosphericPrisonEscape Apr 3 '17 at 4:23
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    $\begingroup$ To prove it, should I write : $$\vec{L} = \vec{constant} = $$ $$\vec{OM} \times \vec{p} = M(r_{i})r_{i}^{2}\dot{\theta}=M(r_{f})r_{f}^{2}\dot{\theta}$$ ?? But how to introduce the two different masses $M_{f}(r)$ or $M(r)$ ? If you could detail the demo. thanks a lot $\endgroup$ – youpilat13 Apr 3 '17 at 4:32
  • $\begingroup$ I'm unsure what you're confused about. You already wrote the identity down with two different masses. $M(r_i)$ will in general be different from $M(r_f)$. And if you then call $M(r_f) = M_f$ is just a matter of notation. $\endgroup$ – AtmosphericPrisonEscape Apr 3 '17 at 4:42
  • $\begingroup$ $M(r)$ is called "the initial NFW mass profile" and $M_{f}(r)$ is called is "the final cumulative mass profile after the disk is formed". Do you think that solution could be found by putting notation $M(r_f) = M_f$ ?? $\endgroup$ – youpilat13 Apr 3 '17 at 4:49
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    $\begingroup$ It's hard to understand the wider picture with just a snippet of your source, but it looks to me like you're trying too hard here. Angular momentum is $L=Mvr$ and angular momentum conservation is just $M_1v_1r_1 = M_2v_2r_2$. If you can say that $v_1=v_2$ (it's unclear to me if that is the case from your snippet) then you're left with $M_1r_1 = M_2r_2$. $\endgroup$ – zephyr Apr 3 '17 at 15:48

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