void eraLd(double bm, double p[3], double q[3], double e[3],
double em, double dlim, double p1[3])
/*
** - - - - - -
** e r a L d
** - - - - - -
**
** Apply light deflection by a solar-system body, as part of
** transforming coordinate direction into natural direction.
**
** Given:
** bm double mass of the gravitating body (solar masses)
** p double[3] direction from observer to source (unit vector)
** q double[3] direction from body to source (unit vector)
** e double[3] direction from body to observer (unit vector)
** em double distance from body to observer (au)
** dlim double deflection limiter (Note 4)
**
** Returned:
** p1 double[3] observer to deflected source (unit vector)
**
** Notes:
**
** 1) The algorithm is based on Expr. (70) in Klioner (2003) and
** Expr. (7.63) in the Explanatory Supplement (Urban & Seidelmann
** 2013), with some rearrangement to minimize the effects of machine
** precision.
**
** 2) The mass parameter bm can, as required, be adjusted in order to
** allow for such effects as quadrupole field.
**
** 3) The barycentric position of the deflecting body should ideally
** correspond to the time of closest approach of the light ray to
** the body.
**
** 4) The deflection limiter parameter dlim is phi^2/2, where phi is
** the angular separation (in radians) between source and body at
** which limiting is applied. As phi shrinks below the chosen
** threshold, the deflection is artificially reduced, reaching zero
** for phi = 0.
**
** 5) The returned vector p1 is not normalized, but the consequential
** departure from unit magnitude is always negligible.
**
** 6) The arguments p and p1 can be the same array.
**
** 7) To accumulate total light deflection taking into account the
** contributions from several bodies, call the present function for
** each body in succession, in decreasing order of distance from the
** observer.
**
** 8) For efficiency, validation is omitted. The supplied vectors must
** be of unit magnitude, and the deflection limiter non-zero and
** positive.
**
** References:
**
** Urban, S. & Seidelmann, P. K. (eds), Explanatory Supplement to
** the Astronomical Almanac, 3rd ed., University Science Books
** (2013).
**
** Klioner, Sergei A., "A practical relativistic model for micro-
** arcsecond astrometry in space", Astr. J. 125, 1580-1597 (2003).
**
** Called:
** eraPdp scalar product of two p-vectors
** eraPxp vector product of two p-vectors
**
** Copyright (C) 2013-2014, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
int i;
double qpe[3], qdqpe, w, eq[3], peq[3];
/* q . (q + e). */
for (i = 0; i < 3; i++) {
qpe[i] = q[i] + e[i];
}
qdqpe = eraPdp(q, qpe);
/* 2 x G x bm / ( em x c^2 x ( q . (q + e) ) ). */
w = bm * ERFA_SRS / em / ERFA_GMAX(qdqpe,dlim);
/* p x (e x q). */
eraPxp(e, q, eq);
eraPxp(p, eq, peq);
/* Apply the deflection. */
for (i = 0; i < 3; i++) {
p1[i] = p[i] + w*peq[i];
}
/* Finished. */
}
void eraRefco(double phpa, double tc, double rh, double wl,
double *refa, double *refb)
/*
** - - - - - - - - -
** e r a R e f c o
** - - - - - - - - -
**
** Determine the constants A and B in the atmospheric refraction model
** dZ = A tan Z + B tan^3 Z.
**
** Z is the "observed" zenith distance (i.e. affected by refraction)
** and dZ is what to add to Z to give the "topocentric" (i.e. in vacuo)
** zenith distance.
**
** Given:
** phpa double pressure at the observer (hPa = millibar)
** tc double ambient temperature at the observer (deg C)
** rh double relative humidity at the observer (range 0-1)
** wl double wavelength (micrometers)
**
** Returned:
** refa double* tan Z coefficient (radians)
** refb double* tan^3 Z coefficient (radians)
**
** Notes:
**
** 1) The model balances speed and accuracy to give good results in
** applications where performance at low altitudes is not paramount.
** Performance is maintained across a range of conditions, and
** applies to both optical/IR and radio.
**
** 2) The model omits the effects of (i) height above sea level (apart
** from the reduced pressure itself), (ii) latitude (i.e. the
** flattening of the Earth), (iii) variations in tropospheric lapse
** rate and (iv) dispersive effects in the radio.
**
** The model was tested using the following range of conditions:
**
** lapse rates 0.0055, 0.0065, 0.0075 deg/meter
** latitudes 0, 25, 50, 75 degrees
** heights 0, 2500, 5000 meters ASL
** pressures mean for height -10% to +5% in steps of 5%
** temperatures -10 deg to +20 deg with respect to 280 deg at SL
** relative humidity 0, 0.5, 1
** wavelengths 0.4, 0.6, ... 2 micron, + radio
** zenith distances 15, 45, 75 degrees
**
** The accuracy with respect to raytracing through a model
** atmosphere was as follows:
**
** worst RMS
**
** optical/IR 62 mas 8 mas
** radio 319 mas 49 mas
**
** For this particular set of conditions:
**
** lapse rate 0.0065 K/meter
** latitude 50 degrees
** sea level
** pressure 1005 mb
** temperature 280.15 K
** humidity 80%
** wavelength 5740 Angstroms
**
** the results were as follows:
**
** ZD raytrace eraRefco Saastamoinen
**
** 10 10.27 10.27 10.27
** 20 21.19 21.20 21.19
** 30 33.61 33.61 33.60
** 40 48.82 48.83 48.81
** 45 58.16 58.18 58.16
** 50 69.28 69.30 69.27
** 55 82.97 82.99 82.95
** 60 100.51 100.54 100.50
** 65 124.23 124.26 124.20
** 70 158.63 158.68 158.61
** 72 177.32 177.37 177.31
** 74 200.35 200.38 200.32
** 76 229.45 229.43 229.42
** 78 267.44 267.29 267.41
** 80 319.13 318.55 319.10
**
** deg arcsec arcsec arcsec
**
** The values for Saastamoinen's formula (which includes terms
** up to tan^5) are taken from Hohenkerk and Sinclair (1985).
**
** 3) A wl value in the range 0-100 selects the optical/IR case and is
** wavelength in micrometers. Any value outside this range selects
** the radio case.
**
** 4) Outlandish input parameters are silently limited to
** mathematically safe values. Zero pressure is permissible, and
** causes zeroes to be returned.
**
** 5) The algorithm draws on several sources, as follows:
**
** a) The formula for the saturation vapour pressure of water as
** a function of temperature and temperature is taken from
** Equations (A4.5-A4.7) of Gill (1982).
**
** b) The formula for the water vapour pressure, given the
** saturation pressure and the relative humidity, is from
** Crane (1976), Equation (2.5.5).
**
** c) The refractivity of air is a function of temperature,
** total pressure, water-vapour pressure and, in the case
** of optical/IR, wavelength. The formulae for the two cases are
** developed from Hohenkerk & Sinclair (1985) and Rueger (2002).
**
** d) The formula for beta, the ratio of the scale height of the
** atmosphere to the geocentric distance of the observer, is
** an adaption of Equation (9) from Stone (1996). The
** adaptations, arrived at empirically, consist of (i) a small
** adjustment to the coefficient and (ii) a humidity term for the
** radio case only.
**
** e) The formulae for the refraction constants as a function of
** n-1 and beta are from Green (1987), Equation (4.31).
**
** References:
**
** Crane, R.K., Meeks, M.L. (ed), "Refraction Effects in the Neutral
** Atmosphere", Methods of Experimental Physics: Astrophysics 12B,
** Academic Press, 1976.
**
** Gill, Adrian E., "Atmosphere-Ocean Dynamics", Academic Press,
** 1982.
**
** Green, R.M., "Spherical Astronomy", Cambridge University Press,
** 1987.
**
** Hohenkerk, C.Y., & Sinclair, A.T., NAO Technical Note No. 63,
** 1985.
**
** Rueger, J.M., "Refractive Index Formulae for Electronic Distance
** Measurement with Radio and Millimetre Waves", in Unisurv Report
** S-68, School of Surveying and Spatial Information Systems,
** University of New South Wales, Sydney, Australia, 2002.
**
** Stone, Ronald C., P.A.S.P. 108, 1051-1058, 1996.
**
** Copyright (C) 2013-2015, NumFOCUS Foundation.
** Derived, with permission, from the SOFA library. See notes at end of file.
*/
{
int optic;
double p, t, r, w, ps, pw, tk, wlsq, gamma, beta;
/* Decide whether optical/IR or radio case: switch at 100 microns. */
optic = ( wl <= 100.0 );
/* Restrict parameters to safe values. */
t = ERFA_GMAX ( tc, -150.0 );
t = ERFA_GMIN ( t, 200.0 );
p = ERFA_GMAX ( phpa, 0.0 );
p = ERFA_GMIN ( p, 10000.0 );
r = ERFA_GMAX ( rh, 0.0 );
r = ERFA_GMIN ( r, 1.0 );
w = ERFA_GMAX ( wl, 0.1 );
w = ERFA_GMIN ( w, 1e6 );
/* Water vapour pressure at the observer. */
if ( p > 0.0 ) {
ps = pow ( 10.0, ( 0.7859 + 0.03477*t ) /
( 1.0 + 0.00412*t ) ) *
( 1.0 + p * ( 4.5e-6 + 6e-10*t*t ) );
pw = r * ps / ( 1.0 - (1.0-r)*ps/p );
} else {
pw = 0.0;
}
/* Refractive index minus 1 at the observer. */
tk = t + 273.15;
if ( optic ) {
wlsq = w * w;
gamma = ( ( 77.53484e-6 +
( 4.39108e-7 + 3.666e-9/wlsq ) / wlsq ) * p
- 11.2684e-6*pw ) / tk;
} else {
gamma = ( 77.6890e-6*p - ( 6.3938e-6 - 0.375463/tk ) * pw ) / tk;
}
/* Formula for beta from Stone, with empirical adjustments. */
beta = 4.4474e-6 * tk;
if ( ! optic ) beta -= 0.0074 * pw * beta;
/* Refraction constants from Green. */
*refa = gamma * ( 1.0 - beta );
*refb = - gamma * ( beta - gamma / 2.0 );
/* Finished. */
}
