Simulation method

The simulations utilised a wind-scatter model of the atmosphere (see e.g. [10]) using two Taylor screens of frozen Kolmogorov turbulence with large ($>1000r_{0}$) outer scale. Fluctuations in air and water vapour spectral dispersion were not modelled, and the Taylor screens introduced equal fluctuations in the wavefront delay at all wavelengths.

In the simulations, the Taylor screens were blown past the simulated telescopes at constant wind velocities. The timescale for changes in the light intensity measured with a large diameter telescope is proportional to the parameter $\left (r_{0}/\Delta v\right )$ (see [10,11]) where $\Delta v$ is the standard deviation of the distribution of the wind velocities for the screens, weighted by the turbulence $C_{N}^{2}$ for each screen:

$$\Delta v = \left [ \frac{ \int_{0}^{\infty} \left \vert \mat... ... \left ( h \right ) \mbox{d}h} \right \vert^{2} \right ]^{1/2}$$ (17)

and using the same definitions for the layer height $h$, the wind velocity $\mathbf{v}$ and the turbulence strength $C_{N}^{2}$ as Roddier ([10]).

The Taylor screens were moved across the telescope aperture at the appropriate wind speeds along one axis of the rectangular Taylor screen array. A section of each Taylor screen was extracted at each time point and then rotated according to the wind direction angle and re-sampled to have at least twice as many pixels per $r_{0}$ using linear interpolation to minimise pixel aliasing. The resulting screens were summed and converted to complex wavefronts at a range of different wavelengths. After truncating the wavefronts with a circular telescope aperture, an image plane representation of the wavefronts was generated at each wavelength using an FFT. The atmospheric model used throughout this documents had two Taylor screens of equal strength moving at equal wind speed with wind angles differing by $120\hbox{$^\circ$}$. The time unit used when describing the simulations corresponds to the time taken for each of the Taylor screens to move by the coherence length $r_{0}$ for the wavefronts in the telescope aperture plane (after the wavefronts had been perturbed by both atmospheric layers).

Seven discrete wavelength channels were simulated, with equal spacings in wavelength between $1.97\mbox{ $\mu$m}$ and $2.43\mbox{ $\mu$m}$. The correlated fluxes in the seven wavelength channels were then linearly combined to give three channels with wavelengths and bandpasses approximately matching the channels on the real PRIMA FSUs.

In some of the simulations, bulk atmospheric refraction and/or scintillation were also modelled. For these studies one of the Taylor screens was assigned an altitude of $500\mbox{ m}$ above the telescope and the other $5000\mbox{ m}$. For refraction studies Snell's law was used to model the variation of the light ray position and tilt with wavelength and altitude, ensuring that the correct part of each Taylor screen was used for each wavelength channel. This was calculated using the approach introduced in Section 2.2.7.

The fluctuation in delay induced by the atmospheric turbulence ( $\Delta z\left( \mathbf{r'}+\mathbf{\Delta r'}\left ( \lambda \right ), t,\lambda \right )$ from Equation 17) has only a small direct dependence on wavelength $\lambda$ (due to dispersion), so the induced phase rotation is well approximated across the K band by:

$$\phi\left( \mathbf{r},t,\lambda \right ) =\frac{\Delta z\lef... ...'}\left ( \lambda \right ),t,\lambda_{cen} \right )}{\lambda}$$ (18)

where $\Delta z\left( \mathbf{r'}+\mathbf{\Delta r'}\left ( \lambda \right ),t,\lambda_{cen} \right )$ is the delay induced by atmospheric fluctuations at the centre of the K band (wavelength $\lambda_{cen}$) at position $\mathbf{r'}+\mathbf{\Delta r'}\left ( \lambda \right )$ and time $t$. This approximation was utilised in all the numerical simulations of the effects of seeing on astrometric performance presented in this document. Note that in accurate calculations of the effect of atmospheric refraction and dispersion these simplifications should not be used.

For the simulations, the air density as a function of altitude was based on the Glenn Research Center's Earth Atmosphere Model, and the refractive index was calculated from the density using a cubic spline fit to K-band data from [12]. The algorithms used are presented in Figures 2 and 3. The optical ray displacement was calculated and integrated at $50\mbox{ m}$ intervals through the atmosphere - Richard Mathar has recently started work on an improved model for the optical path through the atmosphere.

図 2: Algorithm used to estimate air density as a function of altitude for simulations of atmospheric seeing effects

図 3: Algorithm used to estimate refractivity as a function of altitude and wavelength for simulations of atmospheric seeing effects

For the scintillation studies a first-order approximation to the optical propagation was performed. The effect of each Taylor screen was investigated independently, with the amplitude fluctuation ( $\chi_{p} \left(\mathbf{r},t\right)$ from Equation 2) in the AT aperture plane estimated by applying phase changes in the conjugate plane to re-image the Taylor screen to the appropriate altitude. The amplitude fluctuations from the two Taylor screens were then combined multiplicatively without taking account of any second-order terms resulting from the interaction of the wavefront fluctuations induced by the two Taylor screens. The phase changes resulting from optical propagation from the turbulent layers to the telescopes were also ignored.

The spatial sampling of the electric field was kept constant in the image plane for all the simulations which necessarily required wavelength-dependent spatial sampling in the pupil plane.