# Re: velocity units

From: Melanie Jo Clarke <mjdosaj_at_MIT.EDU>
Date: Mon, 30 Oct 2006 12:05:39 -0500

Hello,

My confusion stemmed from several things:
1) Given that the canonical momentum is p = a^2 * dx/dt, I couldn't see
where a
factor of sqrt(a) came in. I didn't know it was there for historical reasons,
so I tried to derive it. Although the equations given in your paper involved
p, it was stated in the manual that the time variable was ln(a), so I
went from
there. In reference to the conversion, the manual only stated that the
velocity
was in internal units.
2) sqrt(a) didn't work for me, so I tried to find something that did.
I should
have asked sooner where the factor came from.

I'm sorry, I misspoke when I mentioned the physical velocity. I meant the
peculiar velocity, of course. Also, the factor I gave was not d(ln a)/dt,
which is H(t), but d(ln a)/d(tau), where d(tau)=Ho*dt/a^2. I should have
composed my thoughts better before sending the email.

Thank you for your careful and patient reply. I am in conversation
with Michele
Trenti, who has generously offered to help me solve my conversion problems.

- Melanie

> Hi Melanie,
>
> The manual is correct on this...
>
> I can only speculate where you got confused, but I'm guessing you have
> mixed up the definitions of comoving, physical, and peculiar velocity.
> Note that they are all different. Also note that the manual says that
> the IC file should contain the *peculiar* velocity divided by sqrt(a),
> not the *physical* velocity as you state above. Let "x" denote comoving
> coordinates and "r=a*x" physical coordinates. Then I call
>
> comoving velocity: dx/dt
> physical velocity: dr/dt = H(a)*r + a*dx/dt
> peculiar velocity: v = a * dx/dt
>
> The physical velocity is hence the peculiar velocity plus the Hubble flow.
>
> I think your attempts to guess the correct conversion factor have caused
> some additional confusion. Let me try to clarify: The internal velocity
> variable of gadget2 is not given by dx/d(ln a). Rather, it is given by
> the canonical momentum p = a^2 * dx/dt, which is different from the
> definition you assumed. The IC-file and snapshot files of gadget don't
> contain the variable "p" directly because of historical reasons of
> compatibility with gagdet-1. Instead, they contain the velocity variable
> u = v/sqrt(a) = sqrt(a) * dx/dt = p / a^(3/2), which is just what the
> manual says. (The conversion between u and p is done on the fly when
> reading or writing snapshot files.)
>
> Also note that d(ln a)/dt is not a*sqrt(omega_m/a + omega_v*a^2 + 1 -
> omega_m - omega_v), as you say above... This factor is equal to the
> Hubble rate, i.e.: d(ln a)/dt = H(a) = H_0 * sqrt(omega_m/a^3 + omega_v
> + (1 - omega_m - omega_v)/a^2).
>
> If you can't solve your unit conversion problem, I'd suggest to talk to
> Michele Trenti, who figured out the grafic->gadget2 conversion.
>
> Best wishes,
> Volker