[mesa-users] Total Energy Integral

[Michael Medford]

Bill,

Thanks as always for your help, and for guiding me to understand the answer
instead of simply having it. I have worked my way through the code and
these definitions.

For total_integral_energy, both dm(k) and energy(k) are averages for the
cell and therefore should be thought of as occurring at the cell's center.
The multiplication of these two factors should yield the internal
energy occurring "within" this zone. Add them all up to get the total
internal energy that occurs between this zone and the surface (where k=1).
Total_internal_energy should be near-zero at the surface and a larger
positive value as the integral penetrates into the star.

For total_gravitational_energy, m_grav(k) and r(k) are defined at the
cell's outer edge. I have checked the profile to ensure that there is no
difference between baryonic and gravitational mass in my models. Indeed
these two values in my circumstances are identical. I see why dm_bar(k) is
the more appropriate term to multiply into m_grav(k) and r(k), since
dm_bar(k) is an average of adjacent zone-centered masses from dm(k) and
ends up lying on each zone's outer edge. Multiplying these factors together
gives you the gravitational energy occurring "at" the outer boundary of the
zone. Add them all up to get the total gravitational energy that occurs
between this zone's outer edge and the surface (where k=1).
Total_gravitational_energy should be near-zero at the surface and a larger
negative value as the integral penetrates into the star.

It seems to me that my volume_term (4*pi*r^2*rho*dr) will always be equal
to dm(k) instead of dm_bar(k), since it only uses values that are defined
as at the center of each cell. (I checked and indeed the volume_term very
nearly equals dm(k). There is a slight discrepancy at the center and
surface due to different boundary conditions, but that should not
dramatically shift the overall shape of the integral). To make sure each
value occurred at the same place in using my volume_term, I shifted each of
my radii and mass values to occur at the center of each cell (I created my
own rmid and averaged the masses between zones). Essentially instead of
shifting a dm(k) to a dm_bar(k) (averaging center values to get outer edge
values), I shifted r(k) and m(k) to the centers (averaging outer edge
values to get center values). By placing both the energy_integral and
gravitational_integral at the center of each zone, I can then add them
together in a simple fashion to get my own total_energy_integral.

It looks to me that the code calculates total_energy_integral without
taking into account the differences between an
energy_integral(k) occurring at the center and a
gravitational_integral(k) occurring at the outer boundary. From
profile_getval.f:

case (p_total_energy_integral)
               val = 0
               do kk=1,k
                  val = val + s% dm(kk)*( &
                     exp_cr(s% lnE(kk)) - &
                     s% cgrav(kk)*s% m_grav(kk)/s% r(kk) + &
                     0.5d0*s% velocity(kk)*s% velocity(kk))
               end do

What am I missing here? It looks like dm(k) is being multiplied into both
the internal energy and the gravitational term (and the kinetic term)
without regard to where the value occurs. I am also unable to find the
eval_section_total_energy_integrals that Richard describes
as occurring in star/private/star_utils.f90. I am running Mesa Version 7393.

Thanks as always for the help!
~Michael

On Mon, Mar 9, 2015 at 6:25 PM, Bill Paxton <paxton at kitp.ucsb.edu> wrote:

> Hi,
>
> I'd suggest a careful reading of the following two lines of code -- think
> about what the various terms mean.  check  where things are defined (cell
> center vs cell boundary).  what's the difference between dm and dm_bar, and
> why do we use dm for total energy and dm_bar for gravitational energy?
>  what's the definition of s% energy(k)?  what disagreements could arise
> between dm or dm_bar vs 4*pi*r^2*rho*dr given that we are considering
> finite volumes rather than differential equations?  hopefully enlightenment
> will follow. ;D
>
> we sum the following for k = 1, s% nz
>          total_internal_energy = dot_product(s% dm(1:k), s% energy(1:k))
>          total_gravitational_energy = -dot_product(s% dm_bar(1:k), &
>             s% cgrav(1:k)*s% m_grav(1:k)/s% r(1:k))
>
> Cheers,
> Bill
>
>
>
> On Mar 9, 2015, at 1:22 PM, Michael Medford wrote:
>
> Hello everyone,
>
> Over the last few weeks I have been investigating the profile column
> variable labelled total_energy_integral. The description of this variable
> is "sum from surface inwards of dm*(v^2/2 - G m / r + E) (ergs)". I was
> confused as to whether this sum includes the necessary volume elements to
> properly sum the spherical shells within this integral. So I set about
> calculating my own energy integral using the necessary data from each cell
> within the profile. Here is how I calculated my integral:
>
> -Routines to shift each of the mass, radii and density terms to the center
> of each cell by taking averages of the outer and inner boundaries
> -Define "dr" as the radii width of each shell
> -Defining an "energy element" = (-Gm/r + E) * (4*pi*r^2*density*dr) where
> all of the terms are in cgs and the velocity of all of the cells are 0
> -Sum from the surface inwards of this energy element
>
> Attached you will see the results of my variable versus the output from
> the profile's total_energy_integral. As the energy integral claims to have
> a dm multiplied into each term, I did not multiply the
> total_energy_integral times anything, since this should be equivalent to
> (4*pi*r^2*density*dr).
>
> Does anyone know why these two values would be different, and by such
> large orders of magnitude? What am I missing about the
> total_energy_integral and what it is recording? Or am I manually summing
> the integral incorrectly?
>
> Thanks so much for the help.
> ~Michael
>
> ----
> Michael Medford
> Northwestern University
>  <Energy_Integrals_2.png><Energy_Integrals_1.png>
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-- 
Michael Medford
Northwestern University
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