The initial conditions

The following parameters are relevant for the initial conditions and can be set in the param.dat file:

  • Input file

  • Input file Format

  • Default rho

  • Angle units

The initial conditions must be provided in a file, and the name of this file must be set with the Input file argument. The file must be a text file and every particle corresponds to a new line in the file. The central mass (Sun) must not be included in the initial conditions. Values for the central mass can be specified directly in the param.dat file, the position and the velocities of the central mass are set to the origin (heliocentric coordinates).

The initial conditions must be a text file and the format must correspond to the values set in the param.dat file (Input file Format: << … >> ). The data of each particle has to be written in a new line in text format, and the format of the data must correspond to the values of Input file Format.

The initial conditions format

The format of the initial conditions file must be specified with the Input file Format option. The different entries must be given between the << and the >> characters and a blank space must be included between every entry. All available options are listed below. Most values are optional, but they must contain a complete set of Cartesian or Keplerian elements, e.g a full set of [x y z vx vy vz] or [a(or P) e inc O w M(orT)]

Possible arguments are:

  • t: start time of the simulation, in years (default = 0.0).

  • i: index of the body. The default value is the line number in the input file, starting with 0. Indices should be unique.

  • m: mass in Solar masses (default = 0.0).

  • r: physical radius in AU. If r is not given or the radius is equal to zero, then the program uses the density to calculate the radius.
    Note that there are different ways to set a radius or density. (See Examples).

  • rho: density in g/cm^3; optional. The default value can be set by the Default rho parameter.

  • x: x-position in AU (heliocentric).

  • y: y-position in AU (heliocentric).

  • z: z-position in AU (heliocentric).

  • vx: x-velocity in AU/day * 0.0172020989 (heliocentric) (See Units).

  • vy: y-velocity in AU/day * 0.0172020989 (heliocentric) (See Units).

  • vz: z-velocity in AU/day * 0.0172020989 (heliocentric) (See Units).

  • a: semi-major axis in AU.

  • P: period in days.

  • e: eccentricity.

  • inc: inclination in radians or degrees (See Angle units).

  • O: (Omega, \(\Omega\)) longitude of the ascending node, in radians or degrees (See Angle units).

  • w: (omega, \(\omega\)) argument of periapsis, in radians or degrees (See Angle units).

  • M: mean anomaly, in radians or degrees (See Angle units).

  • T: Time of first transit, in BJD (See Units).

  • Sx: x-spin in Solar masses AU^2 / day * 0.0172020989. (default = 0.0) (See Units).

  • Sy: y-spin in Solar masses AU^2 / day * 0.0172020989. (default = 0.0) (See Units).

  • Sz: z-spin in Solar masses AU^2 / day * 0.0172020989. (default = 0.0) (See Units).

  • amin: minimal value of semi major axis range for aecount; optional (default = 0.0). (See aeLimits).

  • amax: maximal value of semi major axis range for aecount; optional (default = 100). (See aeLimits).

  • emin: minimal value of eccentricity range for aecount; optional, (default = 0.0). (See aeLimits).

  • emax: maximal value of eccentricity range for aecount; optional, (default = 1.0). (See aeLimits).

  • k2: potential Love number of degree 2, dimensionless (default = 0.0).
    If this is given, then it must also be set in the Output file Format: argument.
  • k2f: fluid Love number of degree 2, dimensionless (default = 0.0).
    If this is given, then it must also be set in the Output file Format: argument.
  • tau: time lag in day / 0.0172020989 (See Units) (default = 0.0).
    If this is given, then it must also be set in the Output file Format: argument.
  • Ic: moment of inertia, dimensionless (See Units) (default = 0.4).
    If this is given, then it must also be set in the Output file Format: argument.

  • Rc: Critical radius for close encounters, rcrit, in AU (default = 0.0)

  • test: Optional value to store in the test arrays.

  • -: skip column, optional.

Examples

Example 1:

format in 'param.dat': << x y z m vx vy vz r >>
input file:
    x1 y1 z1 m1 vx1 vy1 vz1 r1
    x2 y2 z2 m2 vx2 vy2 vz2 r2
    .
    .
    .
    xn yn zn mn vxn vyn vzn rn

GENGA reads the radii from the initial condition file.

Example 2:

format in 'param.dat': << x y z m vx vy vz rho >>
input file:
    x1 y1 z1 m1 vx1 vy1 vz1 rho1
    x2 y2 z2 m2 vx2 vy2 vz2 rho2
    .
    .
    .
    xn yn zn mn vxn vyn vzn rhon

GENGA reads the densities from the initial condition file and computes the radii.

Example 3:

format in 'param.dat': << x y z m vx vy vz >>
Default rho = 2.0
input file:
    x1 y1 z1 m1 vx1 vy1 vz1
    x2 y2 z2 m2 vx2 vy2 vz2
    .
    .
    .
    xn yn zn mn vxn vyn vzn

GENGA uses the default density from the param.dat file and computes the radii.

Example 4:

format in 'param.dat': << x y z m vx vy vz r rho >>
Default rho = 2.0
input file:
    x1 y1 z1 m1 vx1 vy1 vz1 r1 rho1
    x2 y2 z2 m2 vx2 vy2 vz2 r2 rho2
    .
    .
    .
    xn yn zn mn vxn vyn vzn r2 rho2

GENGA reads the radii from the initial condition file. If a radius is set to zero, then GENGA reads the density from the initial condition file and computes the radius.

Units

The units in GENGA are chose such that the solar mass \(M_\odot = 1\), the gravitational constant \(G = 1\) and the distance from the Sun to Earth \(AU = 1\).

With \(G = 6.67408 \cdot 10^-11 \text{m}^3 \text{kg}^{-1} \text{s}^{-2} = 0.01720209895^2 \text{AU}^3 \text{M}_\odot^{-1} \text{day}^{-2}\), and the Gaussian gravitational constant \(k = 0.01720209895\).

Therefore, we need

\[G = 0.01720209895^2 \text{AU}^3 \text{M}_\odot^{-1} \text{day}^{-2} = 1.0 \, \text{day'}^{-2}\]

and

\[1 \, \text{day'} = \text{day} / 0.01720209895,\]

That means that all time units must be rescaled with 0.01720209895.

Time

The time units are given now as \([time] = \text{day} / 0.01720209895 =\) day’.
To convert time from day to day’, it must be multiplicated by 0.01720209895.

Example: 1 min = 0.0006944 day = 0.00001194 day’.

An exception is the time step, it is given in days, and converted internally to code units.

Masses

Masses are given in Solar masses (\(M_\odot\)).

Example: The Sun has a mass of 1 \(M_\odot\).
The Earth has a mass of \(3.003 \cdot 10^{-6} M_\odot\).

Distances, Radii

All distances and radii are given in Astronomical Units, AU.

Example: The distance from the Sun to the Earth is 1 AU.
The Earth radius is \(4.2635 \cdot 10^{-5}\) AU.

Velocities

[v] = AU / day’ = AU / day * 0.01720209895.
To convert velocities from AU / day to AU / day’, they must be divided by 0.01720209895.
Example: The Earth’s orbital velocity is 30 km / s = 0.01720209895 AU / day.
In GENGA units, this is 1.0 AU / day’.

Spin

[spin] = \(M_{\odot} AU^2 / day' = M_{\odot} AU^2 / day \cdot 0.01720209895\)
To convert the spin from \(M_{\odot} AU^2 / day\) to \(M_{\odot} AU^2 / day'\), it must be divided by 0.01720209895.
The spin is computed as
\[\vec{S} = \vec{\Omega} \cdot I,\]

with the angular rotation rate \(\vec{\Omega} = \frac{2 \pi}{P}\) the rotation period \(P\) (in units of day’) and the moment of inertia \(I\):

\[I = I_c m r^2\]

The parameter \(I_c\) defines the inner structure of the body. \(I_c = 2/5\) is a solid sphere with uniform density.

Example: The Sun has a rotational period \(P\) of 27.5 days, a mass of 1.0 \(M_\odot\) and a radius of 0.00464 AU. We assume \(I_c = 0.07\).

\[S = \frac{2 \pi}{27.5 \text{days} \cdot 0.01720209895} \cdot 0.07 \cdot 1.0 M_{\odot} \cdot (0.00464 AU)^2 = 0.00002001703 \, M_{\odot} AU^2 / day'\]

Energy

The energy in the Energy file is reported in in \(M_\odot AU^2 / day^2\).

Angular momentum

The angular momentum in the Energy file is reported in in \(M_\odot AU^2 / day\).

Time lag

The time lag \(\tau\) is given in units of day’ = day / 0.0172020989.
To convert \(\tau\) from day to day’, it must be multiplicated by 0.01720209895.

Example: 1 min = 0.0006944 day = 0.00001194 day’.

The time lag \(\tau\) is related (approximately) to the tidal quality factor \(Q\) via [Efroimsky07]:

\[\tau = \frac{\arctan{(1/Q)}}{2 | n - \omega | },\]

with the mean motion of the planet \(n\) and the rotation rate of the star \(\omega\).

Time of first transit

The time of first transit is given in Barycentric Julian Date (BJD).

Physical constants

The values of used physical constants are set in the define.h file.