The simulation of the experiment has been organized in two main parts. The first one is the simulation of the Sun with all the velocity fields and physical effects that contribute to the solar velocity spectrum, mainly, in the frequency band of interest. The second one is the simulation of the instrumental response from GOLF, including its kinematics respect to the Sun.
In fact, it only consists in calculating what is the intensity measured by
our photomultiplier tubes at any time t. It has to calculate the
convolution between the solar sodium D lines and the reference ones
resulting from the scattering cross section of the sodium in the vapor cell,
shifted by a quantity proportional to the relative line-of-sight velocity.
The result is given as the intensities 
:
where, x and y are the standard solar disk centered rectangular
coordinates, V(x,y) is the velocity component projected onto the line of
sight, c the light speed, 
the central frequency of the line,
is the wavelength relative to the rest wavelength of the
solar line, 
is the Zeeman splitting due to the permanent
magnetic field, 
is the extra shift due to the small magnetic
field provided by a pair of coils, 
is the shape of the line at any point in the solar disk at
anytime, L(x,y) is the limb darkening, and 
is the instrumental response.
Further, the observed velocity by GOLF is the combination of this four intensities in the form of ratios. Amongst others, the classical way (as it is done for actual earth-based instruments) of defining it, would be:
The selected matrix (x,y) to simulate the solar disk has 128
128
pixels with a resolution of 15 arcsec per pixel. These values were chosen as
a good compromise between the computing time and the spatial resolution
needed. However, for some solar phenomena, such as the active regions and
the supergranulation, the calculations are done with a higher order matrix
and we make a "binning'' to reduce the resolution to our standard.
The sodium doublet line profiles have been assumed to be gaussians because
the working points of the instrument are close to its core where these are
well approximated by a gaussian. For the gaussian solar profiles the
parameters introduced for the quiet Sun are a residual intensity of 0.937
and 0.946 and a FWHM of 450 and 650 mÅ for the D1 and D2,
respectively (Boumier 1991). On the other hand, it is well known that
sunspots and active regions produce changes in the profiles integrated over
the solar disk of the solar photospheric absorption lines. In the case of
the sodium doublet, lines become broader, shallower and more asymmetric in
sunspots (Robillot et al. 1993; Ulrich 1992). These line
shapes, when they are added to the quiet Sun line profiles, and integrated
over the solar disk, produce changes in the integrated profile that will be
measured as a Doppler shift giving rise to "fictitious'' velocity fields.
To simulate this behaviour, we have used real data of the sunspot positions
and areas from the Solar Geophysical Data (Boulder). The software devoped is
a generalization of the one built by Jiménez Buendía (1983). It
creates a 256256 matrix of the Sun with a zero value in the pixels
corresponding to the quiet Sun and another one, where there is a sunspot.
The appropriate solar line profile can be selected depending if the pixel
corresponds to a sunspot or a quiet Sun region. For sunspot areas, we have
also kept a gaussian shape with residual intensities of 0.955 and 0.970 and
FWHM of 597.6 and 672.4 mÅ for both D1 and D2 lines. Moreover,
its continuum is 
of the one in the quiet Sun
(Régulo et al.
1993). In Fig. 2 (click here) we can see the velocity
perturbation seen by GOLF for the two considered years. However, there is a
problem because only one image of active regions per day is available in
the bibliographical records. We have steps of one day in the resultant
velocity residuals which have to be corrected for.
As far as the solar continuum intensity hits the cell, it will be only affected by the several filters and optical components that it finds in its way. Changes with time of the properties of these various components will add instrumental noise, but this will not be considered here. However, the change in distance between the spacecraft and the Sun along the year will produce a small modulation of the incident light intensity. In calculating this effect, we found a difference in the incident light of 6.57%.
Figure 2: Velocity perturbation induced by the active regions as
they are seen by the GOLF experiment, corresponding to 1991 in the top, and
1986 in the bottom. The average constant velocity corresponds to the
velocity offset due to the solar gravitational redshift (
)
The velocity fields taken into consideration have been grouped in three
categories: the solar "random'' velocity fields, 
, the solar
deterministic velocity fields, 
, and the kinematics of the
spacecraft relative to the Sun, 
.
The "random'' simulated velocity fields are the ones related to surface
manifestations of the convective zone, such as: supergranulation 
,
giant cells 
. A meridional circulation 
, has also
been added. Their calculated patterns are refreshed every 30 minutes (much
lower than their characteristic time). In this way we have:
The most interesting one in our context, is the supergranulation which has
the highest contribution to the background velocity spectrum at the periods
of hours (Harvey 1985), like the periods predicted for g-modes. There are
other two important convective movements namely, the granulation and the
mesogranulation. The first one, was not simulated because it influences the
solar velocity spectrum at frequencies 
0.5 mHz, a region far away from
the g-modes. Moreover, its small geometrical scale requires a much higher
spatial resolution and, due to its lower characteristic timescale, needs a
much lower refreshing time than the others (increasing enormously the amount
of computing time). However, a constant effect attributed to it, the
velocity limb shift, has been considered and will be described later. The
second one, the mesogranulation, was not simulated because it is a velocity
field not well studied and its influence in the frequency region of interest
for GOLF is believed to be small (Harvey 1985).
To simulate the supergranulation (), giant cells () and
meridional circulation (), a software initially devoped by
Hathaway
(1987, 1988), kindly provided by F. Hill, has been optimized and used. It is
based on a decomposition of the velocity field vector over the solar surface
in its poloidal and toroidal modes spectrum. The calculated matrix has
256256 pixels with a resolution of 7.5 arcsec per pixel. Each point
is also subdivided in 4 to increase the accuracy. The selected refreshed
time of 30 minutes was chosen as a good compromise between the CPU time
spent on each loop and the continuity of the images. The supergranulation is
parametrized by an horizontal scale of 30000 km and a lifetime of 1 day
(Anderson & Avrett 1991). Then, we choose a spectrum of modes between
1
129 with 1 |m| . The individual
amplitudes chosen yielded line-of-sight velocities up to 
410
ms-1. The integration over the solar disk yields residual velocities up
to 
2 ms-1, values that were selected to be consistent with
those observed
(Pallé et al. 1995). The giant cells are convective
structures (not unambiguously detected yet) with predicted sizes of the
order of 100000 km and a lifetime of 30 days. In this case, the modes chosen
were sectorial (m=l) with 
. The integrated residual
velocities are in the range of 
ms-1.The selected modes
to simulate the meridional circulation were those with 
and 
with m=0 and individual velocity amplitudes of ms-1.
This selection is made on the basis of similarity to what it is observed
and/or it is predicted (Hathaway 1988).
Other solar velocity fields that have been well measured and are of
deterministic nature have been also taken into account. These are:
where,
, is the projected solar differential rotation, which has
been taken from the spectroscopically measured expression
(Howard & Harvey
1970):
, is the gravitational red-shift. It represents the
velocity equivalent of the difference of gravitational field between the
sodium atoms at the Sun's photosphere and the experiment's cell at L1. It
has a constant value of 
ms-1 and it is important because
it shifts the position on the solar line, where the measurements are made.
, is the so-called velocity limb-shift, that appears
when the mean line position is measured as a function of position on the
Sun. It is found that the convective blue shift decreases from the center
towards the limb of the solar disc (Schröter 1957;
Beckers & Nelson
1978). A numerical approximation can be used:
where k1 and k2 depend on the observed line. Unfortunately, there is
no measured values available for the sodium lines. Instead of that, we have
used k1= 125 and k2= 2, which are those measured for the potassium
resonance line.
, is the solar oscillations velocity contribution.
Earth-based solar disk integrated spectrophotometers can measure those
modes with 
. We have simulated both p and g-modes up to these
degrees. For the acoustic ones, we have introduced 22 different modes with
for each value of , with measured amplitude levs
(Régulo, 1987). For the gravity waves, the selected ones were for the
, 
; for , 
; and for
, 
, with 1 mms-1 equal amplitude, as
predicted by most of the theories.
The kinematic effects of the spacecraft can be calculated as:
where, 
, is the radial orbital velocity of the spacecraft at
L1. It can be calculated as 
,
where 
is the angle of the Sun-spacecraft vector at any day with
respect to its origin at the winter solstice, obtained by inverting the
equation:
where e is the eccentricity of the orbit. For this velocity field we
calculate one value every day and then we make a linear interpolation to get
it anytime during the day.
, is the orbit of the spacecraft around the
Lagrangian L1 point. The maximum halo velocity is expected to be 
ms-1 with a six months period. Therefore we have used a simple sine
expression with this period and amplitude.
The scattering cross section of the vapor cell will give the reference lines
and due to the special magnetic configuration of the instrument, only the
components will be excited giving three groups, D1, D2a and
D2b. Each one has four hyperfine components. The relative heights are in proportion 2:1:3 respectively. The FWHM of the hyperfine components are 18.5
(Boumier 1991). The Zeeman components have been shifted, from its original position, by a quantity of 106.1, 132.3 and
79.3 mÅ due to the permanent longitudinal magnetic field of 5000 Gauss. The separation among the hyperfine components are 5.6, 5.3, 4.8 mÅ, for D1, D2a and D2b, respectively.
Moreover, the extra is due to the modulating magnetic field produced by a pair of coils in the pole pieces of the permanent magnet. This has the amplitude of 
100 Gauss allowing an instantaneous calibration of the instrument.
