ASTRONOMICAL REFRACTION APPLIED RESEARCH Astronomical Refraction Michael E. Thomas and Richard I. Joseph A stronomical observations near the horizon have historically been made for naval navigation and for determining the time of sunset. Recent applications include near-horizon daytime scene simulation to support infrared search and track and infrared seeker studies. The ray path of the setting or rising Sun can be highly distorted near the horizon, and it is seldom rectilinear. An accurate model of atmospheric refraction must include the observer’s altitude as well as range, frequency, and atmospheric pressure and temperature. At infrared through ultraviolet frequencies, refractivity depends strongly on the vertical temperature profile, and an explicit relationship between refractivity and temperature can be obtained. This article describes a refraction model we developed that closely reproduces the profile of the setting Sun. INTRODUCTION The real part of the atmospheric index of refraction observations as a function of the inclination angle in is a function of pressure, temperature, and frequency. the visible/infrared portion of the electromagnetic spec- Many interesting low-altitude refractive effects exist trum. The model is applied to observations of sunset. because of tropospheric variations in density and water- vapor partial pressure as a function of position. Atmo- spheric refraction is divided into three categories: as- REFRACTIVITY OF A STANDARD tronomical, terrestrial, and geodesic. Astronomical ATMOSPHERE refraction addresses ray-bending effects for objects out- side the Earth’s atmosphere relative to an observer Because of the abundance of nitrogen and oxygen, within the atmosphere. Terrestrial refraction considers the contributions of those elements dominate the re- the case when both object and observer are within the fractivity of a dry atmosphere. Nitrogen and oxygen Earth’s atmosphere. Geodesic refraction is a special molecules have no infrared bands of importance to the case of terrestrial refraction where the object and refractive index, so that only electronic bands need to be considered for a model valid from millimeter waves observer are at low altitudes, as is commonly the case 1 in surveying. to the ultraviolet. On the basis of the work of Edlen, Two components of a complete atmospheric refrac- a simple semi-empirical model for the dry air refractiv- tion model are (1) a representation of the index of ity, Ndry, is given by refraction of the atmosphere as a function of pressure, temperature, and frequency, and (2) a description of =−×6 Nndry() dry 110 the ray path. This article presents an improved general (1) − numerical procedure for computing astronomical =+×(.776 2 4 . 36 10 82n )PT /, dry JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 17, NUMBER 3 (1996) 279 M. E. THOMAS AND R. I. JOSEPH where ndry is the dry air index of refraction, n is the all higher altitudes. The representation of refractivity as wavenumber (reciprocal of the wavelength) in cm21, a function of altitude in Eq. 2 is a simple yet useful Pdry is the total dry air pressure in kilopascals, and T relationship for ray path modeling. is the temperature in kelvins. This model is valid at altitudes where the mixing ratio between oxygen and nitrogen is fixed (<100 km) and from 0.2 to 2,000 mm RAY PATH MODEL (5 to 50,000 cm21); it also compares well with other Because the index of refraction depends on density, models.2,3 and the density of the atmosphere strongly depends on By using the hydrostatic equation for total pressure, altitude, light propagating in the atmosphere is bent and given the temperature profile, the altitude depen- (typically toward lower altitudes or regions of higher dence z9 can also be included in the dry air refractivity density). The density of the atmosphere does not vary as follows: greatly in the horizontal direction and, hence, only the vertical structure needs to be considered. For an observ- er at altitude z90 viewing an astronomical object at an ′×−82 NTz))=.+.dry(,nn ( (776 2 4 36 10 ) altitude z9 above a spherically stratified Earth as ≥ z′ illustrated in Fig. 1, the refracted path u(z9, d) for d 0 ′(2) 4 Pzdry()0 mg ⌠ dz′′ is given by ×exp − , ′ ⌡ ′′ Tz)( kB Tz( ) z′ 0 z′ ⌠ ′′ ′ = dz (3) where z90 is the observer altitude above sea level, m is ud()z, , the average molecular mass of a dry atmosphere, g is [nz (′′ )( R+ z′′ )]2 ⌡ ()Rz+ ′′ E sec2d− 1 E 2 gravitational acceleration, and kB is Boltzmann’s con- ′ ′ + ′ z0 [nz (00 )( RE z )] stant. For altitudes up to 100 km, mg/kB is 34.16 K/km. The vertical temperature profile for the 1976 U.S. Stan- dard Atmosphere has a lapse rate in the troposphere of where n(z″) is the index of refraction as a function of 26.5 K/km (first 11 km) and a constant temperature in altitude, RE is the Earth’s radius, and d is the inclination the stratosphere (11 to 20 km). Above 20 km, the index (or elevation) angle. Equation 3 has been applied by of refraction is close to 1, and details of the temperature astronomers and navigators for many years to correct for dependence become less important. For this reason, the atmospheric refraction. An approximate solution has constant temperature of the stratosphere is applied to been obtained by Garfinkel5 for a standard atmosphere. Actual ray path ddev Apparent line Source of sight z09 R 9 = R + z 9 d dunrefr E uunrefr u RE Earth Figure 1. Illustration of the angle of deviation for viewing an astronomical object by an observer at altitude z 09 and inclination angle d within a spherically stratified atmosphere surrounding the Earth. u, refracted ray path; uunrefr, unrefracted ray path. 280 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 17, NUMBER 3 (1996) ASTRONOMICAL REFRACTION For general index of refraction vertical profiles, howev- er, numerical integration techniques are necessary to 200 pddf(Z′′( z , ), ) ud()z,′ = ∑ i1 obtain the refracted path. Simpson’s rule is commonly i=1 200 applied (one important user of this approach is the U.S. 6 −Rz+′ Naval Observatory). The vertical temperature profile +sin1 E 0 nz (′ )cosd (5) +′ 0 is often constructed with a piecewise continuous rep- RzE 1 resentation, which produces cusps at boundaries be- − Rz+′ tween layers. Unfortunately, numerical instability prob- −sin1 E 1 cosd , Rz+′ lems occur at these cusps in the vertical temperature E profile. This representation also limits the general ap- plicability of the method since the integrand must be segmented between the locations of the cusps. A rea- where the first argument of the function f is defined as sonably efficient and stable numerical approach is to apply Gauss-Chebyshev quadrature numerical integra- tion to the preceding integral. A simple variable sub- cosdd+Zz (′ , ) stitution leads to the following equivalent form Zz,′′()d = 1 for Eq. 3: i 1 2 cosdd−Zz (′ , ) ()21 i− + 1cos p, 2 400 cosd ⌠ 1 udd(Zz, (′ ), ) = ⌡ − ′′′ 2 1 Z(z,d) and f is given by Zz(,)′ d (4) dZ′ × , dn(Z′′′ (z ,dd )) C() + 1 dz′′Z(z, ′ ′′ d ) [ZZz′− (′ ,dd )](cos− Z′ ) ′ = ii1 fZ,()i d −′2 1Zi where the following substitutions have been made: 1 × . dn(()) z′′ Z ′ C ()d + 1 ′+′ ′′ ′ nz()(00 RE z ) dz ′ Zi C()d =, Zi secd nz()(′ R+ z′ ) Zz,()′dd=00E cos , ′+′ nz()( RE z ) Of particular interest is the computation of the deviation angle ddev as a function of the inclination angle d, as defined in Fig. 1. The unrefracted ray path nz()(′ R+ z′ ) Zz,′′′()dd= 00E cos . u can be determined exactly from Eq. 3 for n con- ′′ + ′′ unrefr nz()( RE z ) stant. The result is This form of the integrand also points out an inte- grable singularity in the leading factor for the important −−Rz+′ ud(z,′ )=− sin11 [cos( d )] sinE 0 cos( d ) .(6) case when d = 0. Although not a problem for an exact unrefr +′ RzE calculation, it can be a problem for a numerical calcu- lation. Because the leading factor is the weighting func- tion for Chebyshev polynomials, the chosen numerical The difference between the unrefracted and refracted approach avoids the singularity. Furthermore, the pre- inclination angles determines the deviation angle, ceding integral can be solved exactly when the index ddev(d). It is given in terms of unrefracted and refracted of refraction is constant, which is essentially the case u angles by above the Earth’s atmosphere (n = 1). Within the at- mosphere, a numerical approach is needed. The imple- mentation of a 200 term Gauss-Chebyshev quadrature ′ = ′ − ′ ddududdev(z0 , ) unrefr ()() z, z, from z90 to z91 (= 100 km), and the exact solution for Rz+ ′ (7) constant index from 100 km to z9, lead to the following + E 0[sin(uu )− sin( )]. approximate form: Rz+ ′ unrefr E JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 17, NUMBER 3 (1996) 281 M. E. THOMAS AND R. I. JOSEPH RESULTS about 130 m. The temperature at the observer was ° We checked the results given by Eq. 7 by computing about 31 C with a lapse rate of 28 K/km. the angular deviation of an astronomical object ob- Another example with the Sun at a higher elevation served through a standard atmosphere as a function of angle and the observer at a lower altitude is shown in Fig. 3, which compares the observed rim of the setting elevation angle. The temperature profile is that of the 8 1976 U.S. Standard Atmosphere as described earlier Sun in a photograph from Greenler’s book with the and is used in conjunction with Eq. 2 to provide a computed shape of the Sun. A standard atmosphere is model of the vertical refractivity. Using this profile, we used in this case with an observer at 10 m.