Oriani's theorem derived

Introduction

Oriani's theorem deals with a coefficient in the Lambert's series expansion of astronomical refraction in powers of tan Z, where Z is the apparent zenith distance at the observer. Although Oriani himself found a rather inconvenient proof, a much more elegant one was discovered by E. T. Whittaker and published in Sir Robert Ball's “Treatise on Spherical Astronomy,” pp. 123–124. As this reference is no longer readily available, the derivation is reproduced here, in a slightly simpler notation. (I have followed a similar line in my tutorial paper (2006) on atmospheric refraction.)

Notation

The Web isn't very friendly to mathematics. It's awkward to set subscripts and superscripts and long fractions. Browser support for MathML is still very uneven. So I'll do the best I can here with plain HTML.

To avoid a lot of subscripts, I'll use upper-case (“capital”) letters for the values of variables at the observer. That makes them parameters of the problem. The lower-case letters will be used for the variables themselves.

In printed works, mathematical symbols are usually set in italic type. That looks a little pale on the computer screen, so I'm using boldface here instead, in the running text. Equations that are set off by themselves don't need this enhancement, and will be set in the normal face.

The integrand

Remember that the refraction integral is refr = ∫_N¹ −tan z dn/n

= ∫₁^N tan z dn/n ,
where the value of n, the refractive index, is N at the observer, and 1 at the top of the atmosphere.

Eliminating z

The development of the refraction integral as a series expansion begins by writing the integrand, tan z dn/n, in terms of the apparent zenith distance Z, instead of the local value z along the refracted ray.

If we write R for the observer's distance from the center of curvature of the atmosphere, and (R+h) for the local radius at some point along the ray, the refractive invariant is

I = NR sin Z = n(R+h) sin z

at every point along the ray, where N is the value of the refractive index n at the observer. (Recall that I'm using upper-case symbols for the values of n and z at the observer, to avoid using subscripts.)

Then, at an arbitrary point along the ray, sin z is just I/n(R+h); or, in terms of the values at the observer,

sin z = (NR sin Z) / [n(R+h)] .

Unfortunately, this becomes rather messy when substituted into tan z = sin z/cos z, because we have to use sqrt(1−sin²z) for cos z in the denominator:

tan z = { (NR sin Z) / [n(R+h)] } / sqrt {1 − [(NR sin Z) / n(R+h)]²} .

Change of variable

This becomes somewhat more manageable if we make the substitution

(R + h)/R = 1 + s ,
so that s is just height measured in Earth radii. Note that s is always small: it is 0.01 at 64 km height, and we can always neglect the refraction above s = 0.02.

So, set

R + h = (1 + s) R
in the refraction integral, and keep only the first-order terms in s, so that

[n (R+h)]² ≈ (1 + 2s) (n R)² .

We can then cancel factors of R in both numerator and denominator of the expression for tan z. The refraction integrand, (tan z) dn/n, becomes

(N sin Z) dn / {n ⋅ sqrt [n² + 2n²s − (N sin Z)² ] } .

The argument of the square root can be rearranged as

(n² − N²sin² Z) + 2n²s = (n² − N²sin² Z) ⋅ [ 1 + 2n²s / (n² − N²sin² Z) ] ;
then the whole refraction integrand becomes

[ (N sin Z) dn / n sqrt (n² − N²sin² Z) ] ⋅ [ 1 + 2n²s / (n² − N²sin² Z) ]^−1/2 .

This looks messy. But what we have done is to produce a fairly simple function of n (the left expression in square brackets), multiplied by a less well-behaved function (the right-hand square bracket) of a form amenable to binomial expansion. We'll expand that latter function and integrate termwise to get something reasonable at the end.

A binomial-series expansion

Now, expand the expression in square brackets on the right, using the binomial theorem; keep only the first-order term in s, and integrate the resulting terms separately, from n = 1 at the top of the atmosphere to n = N at the observer:

refr = ∫₁^N (N sin Z) dn / n sqrt (n² − N²sin² Z) − ∫₁^N (sn N sin Z) dn / [ (n² − N²sin² Z) ]^−3/2 .
The first term is an elementary integral, whose value arcsin (N sin Z) − Z
is exactly the refraction for the plane-parallel atmosphere. Thus the second integral can be regarded as the first-order correction for atmospheric curvature.

This correction term can be evaluated by setting both n and N to unity in its integrand. (The error made is of higher order, as this term is already of order s.) Then the correction term becomes

(sin Z / cos³ Z) ⋅ ∫₁^N s dn .

A physical approximation

Next, we invoke the Gladstone-Dale rule that the refractivity (n − 1) is proportional to the density, ρ. But if (n − 1) = c ρ, then dn = c dρ. This converts the correction term to −c (sin Z / cos³ Z) ⋅ ∫₀^ρ_obs s⋅dρ ,
where ρ_obs is the density at the observer.

The clever part

Finally, integration by parts converts the integral of s dρ to an integral of ρ ds , making the correction

−c (sin Z / cos³ Z) ⋅ ∫₀^s_max ρ⋅ds ,
where the upper limit s_max is the largest normalized height that contributes appreciably to the refraction — essentially, the top of the atmosphere. But this integral of density through the whole atmosphere is just the mass of a unit column; so this last integral is proportional to the surface pressure at the observer, or to the ratio H/R, where H is the scale height or reduced height of the atmosphere.

Furthermore, the factor sin Z / cos³Z = tan Z sec²Z; and if we replace sec² with (1 + tan²), this factor is just (tan Z + tan³Z). The plane-parallel term can also be expressed as a sum of tangent and tangent-cubed terms, if we expand its arcsine in a Taylor series and neglect higher powers of the refractivity. So the sum of the two terms is of the familiar form

refr = A tan Z − B tan³Z ,
where the coefficients A and B involve only conditions at the observer and are independent of the density distribution.

Oriani himself stated that “This expression depends on no hypothesis about either the law of heat in the atmosphere or about the density of the air at various distances from the surface of the Earth.” Laplace provided a more rigorous and complete proof of Oriani's theorem.

Limitations

Originally, Oriani applied the theorem to the first two terms of Lambert's power series in tan Z. The next term, in tan⁵Z, does depend on atmospheric structure — in fact, on the average density gradient.

Obviously, these power series in tan Z must diverge at the horizon. However, they in fact diverge everywhere! It seems to have first been understood by James Ivory that this is only a “semi-convergent” series: the terms get smaller for a while, and then grow bigger and bigger without limit, even near the zenith. But the smallest term in the series is small enough that this approach is useful for numerical purposes out to about 80° zenith distance.

The divergence of the series emphasizes that Oriani's result is restricted to small and moderate zenith distances. Physically, it applies only to the extent that the ray can be regarded as nearly a straight line; that is, the path of the ray through the atmosphere is not appreciably altered by refractive bending. And indeed, the region of validity is where the refraction is no more than a few minutes of arc.

Near the horizon, the refractive ray curvature is comparable to the Earth's curvature. Then, contrary to Oriani's result, the refraction is extremely sensitive to the details of atmospheric structure.

Another problem occurs when the model atmosphere contains water vapor. The mixing ratio of water vapor is much lower in the upper troposphere and the stratosphere than near the ground; but, as the refractivity of water differs from that of dry air, this means that the “constant” in the Gladstone-Dale formula isn't really constant — and so dn cannot be converted to c dρ, which was the trick used above to make integration by parts possible.

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