# Velocity Residuals

After subtracting the fit of the ephemeris velocities to the diode velocities from the diode velocities, some residual signal remains. An example of this residual signal appears in Figure 1. The residual signal typically lies within ± 100 m/s.

The residual signal has several components. The ephemeris calculations do not account for the influence of the Moon and the planets on the Sun-Earth velocity, which can amount to as much as ~30 m/s. The error in the diode velocities is ~6 m/s. These effects combined, however, cannot account for the observed residuals. We suspect that the largest component is due to changes in the pressure and humidity in the interferometer. We suspect that the epoxy seal on the interferometer housing acts as a sieve across which there is a slow diffusion of air and water vapor.

The interferometer has a glass arm and an air arm. The path length through the glass arm ought to be constant (provided the oven temperature remains constant to within specification), but the path length through the air arm changes as the index of refraction of the air changes. The shift in the zero point of the velocity, "dv", due to the change in the index of refraction of the air is

dv = (x_air/x0) c dn

where:

"x_air" represents the thickness of the air arm
"x0" represents the path length difference between the glass arm and the air arm at "standard oven conditions" (35 degrees C, 760 mm Hg, 0% humidity)
"c" represents the speed of light
"dn" represents the change in the index of refraction of air from standard oven conditions.

For the sake of being complete on this point,

x0 = 2 ((n_air*x_air)-(n_glass*x_glass))

where:

"n_air" represents the index of refraction of air
"n_glass" represents the index of refraction of glass
"x_air" represents the path length in the air arm
"x_glass" represents the path length of the glass arm.

The pressure and humidity inside the GONG shelter (outside the interferometer cell), is recorded at one minute intervals at all of the GONG sites. We have used a simple diffusion model to fit the pressure and humidity data for each site to the velocity residuals. The pressure and humidity inside the interferometer cell at the time of each velocity measurement are calculated according to the following equations,

P_in[t(n)] = P_in[t(n-1)] + ((P_out[t(n-1)]-P_in[t(n-1)]) * c_P * (t(n)-t(n-1)))

H_in[t(n)] = H_in[t(n-1)] + ((H_out[t(n-1)]-H_in[t(n-1)]) * c_H * (t(n)-t(n-1)))

where:

"X_in" represents the pressure (X=P) or humidity (X=H) inside the interferometer cell
"X_out" represents the pressure (X=P) or humidity (X=H) outside the interferometer cell
"t(n)" represents the time of the current velocity measurement
"t(n-1)" represents the time of the previous velocity measurement
"c_X" represents the diffusion time constant (X=P for pressure; X=H for humidity)

The index of refraction of the air inside the interferometer cell at the time of each velocity measurement is calculated using Ciddor's equations at the operating wavelength of the instrument, 676.8 nm (Ciddor, P.E. "Refractive Index of Air: New Equations for the Visible and Near Infrared." Applied Optics 35(9):156-1573 (1996)). The diffusion time constants are the free parameters of the fit.

We have found that we achieve much better fits to the 24-hour averages of the pressure and humidity than to the instantaneous pressure and humidity measurements taken at the time of the diode velocity measurements. An example of one of the fits appears in Figure 2. (The gray points in Figure 2 represent the shifts in the velocity zero point due to the chaning index of refraction.) The fit is not exceptional, but the changes in the index of refraction result in changes in the velocity zero point that are of the same magnitude as the velocity residuals and the trends are the same. The diffusion time constants are much longer for air than water, c_P ~ 0.0010 whereas c_H ~ 0.10.

We suspect that the poor quality of the fit is the result of three factors. (1) The air in the interferometer cell is saturated with immersion oil, which has an effect on the index of refraction that we cannot calculate. (2) The barometers and hygrometers at the GONG sites are not well calibrated. (3) The path length differences in the interferometer are not constant across the field of view of the interferometer due to a relative tilt of the mirrors in the arms of the interferometer.

A precise model of the effect of the immersion oil on the index of refraction and calibrated weather instruments and careful accounting of the position of the Sun relative to the interferometer should result in a better fit of the data, but this is not worth the effort. The fact of the matter is that we have arrived at a reasonable explanation for the velocity residuals.

 Jeff Sudol |