Description of Excel program: Grotc.xls and Grotc1.xls

The purpose of the Excel programs is to calculate your own galaxy rotation curve as a function of a bulge, a disk and a halo of darkmatter. This halo is an option
This procedure goes in two steps: There are two dark matter profiles considered: NFW and Hernquist
For a copy of the programs in zip format select grotc.xls.zip This file also contains the programs Grotcexp.xls and Grotcesxp1.xls. For more details of both See: grotcexp.xls
The main physical assumption is that visible matter and darkmatter behave similar. In this simulation for both visible matter and dark matter this is Newton's Law
The main problem or chalenge is not to include to much darkmatter, at least such that the darkmatter density should not be higher than the visible matter, because otherwise it will become visible.

In Excel the distance of the galaxy rotation curve is expressed in Units. One unit is identical with one line of information (results) and is equivalent with 1000 Lightyears.
The average radius of a rotation curve is 100 units or 100000 Lightyears or 30 kpc.
The standard distance of the bulge is 6 units or 6000 Lightyears. This distance is defined by the parameter rbulge. Within this range the speed of the rotation curve increases lineair to 250 km/sec. There after the speed is calculated as a function of the density.
The program consists of approximate 6 tests: Test1, Test2, Test3, Test4, Test5 and Test6
Each test has 11 parameters: "NFW", "r bulge", "r disk", "rdisp", "rs" , "rhos", "a", "Mh", "rho disk str", "rho disk end" and "Filter"

  • The variable "NFW" (or "Hernquist") is used to select the NFW profile. When NFW is 1 the NFW profile is selected. When NFW is 0 the Hernquist profile is selected. When this is the case the description changes into "Hernquist" and the value into 1. When Hernquist is 0 the reverse happens.
  • The variable "r bulge" shows the length of the bulge in Lightyears. The maximum value is 100000.
  • The variable "r disk" shows the length of the disk in Lightyears. The maximum value is 400000.
  • The variable "r disp" shows the length of the disk in Lightyears. The maximum value is 400000.
  • "rs" and "rhos" are two parameters of the NFW dark mattter profile. The dimension of rs is in Lightyears. See http://lanl.arXiv.org/pdf/astro-ph/0301144 page 17 for more details.
  • "a" and "Mh" are two parameters of the Hernquist dark mattter profile. The dimension of "a" is in Lightyears. The dimension of Mh is in 10^12*m0, with m0 being the mass of the Sun. See http://lanl.arXiv.org/pdf/astro-ph/0506015 page 2 for more details.
  • "rho disk start", "rho disk end" and "filter" are input parameters to calculate the disk profile. See below

    In order to start the simulation select START button.

    Each simulation consists only of 1 phase. In effect this phase is the same as phase 4 of the Excel programs Circ11.xls, Circ12.xls etc.

    In this phase the galaxy rotation curve is calculated as a function of the density of the bulge, the disk and the density of dark matter profile (either NFW or Hernquist).
    In the programs Circ11, Circ12 etc the density profile of the disk was calculated based starting from a flat rotation curve (if VSEL = 1). For a description of that program select circ11.xls.htm
    In this program the density is calculated based on the three parameters:

    rhodisks (rho disk start), rhodiske (rho disk end) and filter
    • When r <= rbulge then rho = (v/rbulge)^2 * (3*G/4*pi) with v(rbulge) = 250 km/sec
      In this range rho is constant because v(r)/r is also constant.
      When rhodisks = 0 then rhodisks = rho.
    • when r > rbulge with the following formula :
      rhob=rhodisks - rhodiske
      rho = rhob * filter / (filter + r - rbulge - n) - rhob * filter / (filter + rdisk - rbulge - n) + rhodiske
      The parameter n is calculated such that when r is equal to rbulge + 1, rho is equal to rhodisks. For an exponential disk shape See: grotcexp.xls

    For readers not familiar with rotation curves select this overview article by Stephen M. Kent: http://adsbit.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1987AJ.....93..816K
    There is also a PDF version available from that link. Rememember that Andromeda galaxy, M31 and N224 are one and the same.


  • Program Grotc.xls - NFW Profile

    Test1, Test2 and Test3

    Those 3 tests belong together. There is no dark matter included. The only difference is the parameter "rho disk start" i.e. the density value of the disk immediate when the bulge ends and the disk starts. In order to make the rotation curve smoothly it is assumed that there is also a disk mass inside the bulge.

    Test4, Test5, Test6 and Test7

    Those four tests are used to find the amount of dark matter described by the NFW profile necessary to support a flat galaxy rotation curve . They belong together because the "rho disk start" values are the same (2E-11)
    Test 7 - NFW profile

    Parameters of Excel program Grotc.xls

    Parameter Test1 Test2 Test3 Test4 Test5 Test6 Test7
    1 NFW 1 1 1 1 1 1 1
    2 rbulge 4000 4000 4000 4000 4000 4000 4000
    3 disk 100000 100000 100000 100000 100000 100000 100000
    4 NFW rs 0 0 0 7500 15000 30000 48000
    5 NFW rhos 0 0 0 60E-12 15E-12 3,5E-12 1,35E-12
    6 Hq a 0 0 0 0 0 0 0
    7 Hq MHalo 0 0 0 0 0 0 0
    8 Rho disk s 15.6E-11 12E-11 6E-11 2E-11 2E-11 2E-11 2E-11
    9 Rho disk e 0,0 0,0 0,0 0,0 0,0 0,0 0,0
    10 filter 15 15 15 15 15 15 15
    11 Rho NFW 0 0 0 150E-13 25E-13 8,75E-13 3,375E-13
    12 tot m bulge 3,54 3,54 3,54 3,54 3,54 3,54 3,54
    13 tot m disk 39,57 30,4 15,2 5,07 5,07 5,07 5,07
    14 tot m dm 0.0 0.0 0.0 274,01 506,63 943,19 1490,5
    15 max r dm 0 0 0 291714K 538547K 1005284K 1588234K

    Program Grotc1.xls - Hernquist Profile

    Test1, Test2 and Test3

    Those 3 tests belong together. There is no dark matter included. The only difference is the parameter "rho disk start" i.e. the density value of the disk immediate when the bulge ends and the disk starts. In order to make the rotation curve smoothly it is assumed that there is also a disk mass inside the bulge.

    Test4, Test5, Test6, and Test7

    Those four tests are used to find the amount of dark matter described by the Hernquist profile necessary to support a flat galaxy rotation curve . They belong together because the "rho disk start" values are the same (2E-11)

    Test 7 - Hernquist profile

    Parameters of Excel programs Grotc1.xls

    Parameter Test1 Test2 Test3 Test4 Test5 Test6 Test7
    1 Hernquist 1 1 1 1 1 1 1
    2 rbulge 4000 4000 4000 4000 4000 4000 4000
    3 rdisk 100000 100000 100000 100000 100000 100000 100000
    4 NFW rs 0 0 0 0 0 0 0
    5 NFW rhos 0 0 0 0 0 0 0
    6 Hq a 0 0 0 15000 30000 60000 96000
    7 Hq MHalo 0 0 0 0,27 0,5 0,9 1,35
    8 Rho disk s 15,64E-11 12E-11 6E-11 2E-11 2E-11 2E-11 2E-11
    9 Rho disk e 0,0 0,0 0,0 0,0 0,0 0,0 0,0
    10 filter 15 15 15 15 15 15 15
    11 Rho NFW 0 0 0 0 0 0 0
    12 tot m bulge 3,55 3,55 3,55 3,55 3,55 3,55 3,55
    13 tot m disk 39,6 30,4 15,2 5,07 5,07 5,07 5,07
    14 tot m dm 0.0 0.0 0.0 45 95,1 175 265
    15 max r dm 0.0 0.0 0.0 3850K 7701K 14607K 22624K


    Discussion

    Test3 and Test4 of Grotc.xls and Test2 and Test3 of Grotc1.xls are all examples that dark matter is included.
    What those tests tell you is Test1 and Test7 of Grotc.xls are very interesting from the point of view of visible matter. The same for Test6 and Test7 of Grotc1. Test7 of Grotc.xls and Test7 of Grotc1.xls are the same.


    Created: 23 November 2005

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