Deflection angle and flux

Titan’s stratospheric winds

4.4 Model

4.4.3 Deflection angle and flux

With the coordinate systems defined, we write the basic equations for the bending angle Θ and the normalized flux Φ of a refracted beam of starlight. A ray of starlight penetrating an atmosphere along a line defined by cylindrical coordinates (ρ, θ) will experience a bending angle Θ(ρ, θ) (bold font denotes a vector quantity, in this case with components in the ˆρ and ˆθdirections) given by the gradient of the refractivity ν(r, φ) integrated along the path of the ray. Since the bending angle in the present situation is always small (Θ < 0.⁰⁰9), the path of the integration can to a very good approximation be considered a straight line parallel to thez axis,

Θ(ρ, θ) =∇^{Z ∞}

−∞ν(r, φ)dz. (4.9)

The complexity of the case of a non-spherical atmosphere tilted with respect to the line of sight arises from the range of latitudes φ and the varying plane of refraction Θ/|Θ| experienced by each ray.

The observed flux of starlight which has penetrated Titan’s atmosphere is a function of three competing factors. These can be expressed simply for a spherically symmetric atmosphere, as follows (EY92). The spreading of a bundle of rays by differential refraction as it traverses the atmosphere will lead to a reduction of the flux by a factor ofdρ/dρ⁰. The flux may be further reduced by absorption or scattering in Titan’s atmosphere, which can be expressed as the exponential of the negative of the total opacity integrated along the path of the ray τ_obs(ρ). Finally, the curvature of Titan’s limb in the plane perpendicular to the starlight will cause rays to converge within the shadow, enhancing the flux by a factor proportional to the radius of curvature divided by the distance of the observer from the center of curvatureρ/ρ⁰. Combining these terms, the flux of starlight emerging from a spherically symmetric atmosphere, normalized to that incident, will be

Φ(ρ) = ρ ρ⁰

dρ

dρ⁰ exp[−τobs(ρ)]. (4.10)

In the absence of spherical symmetry, it is perhaps more intuitive to think of the normalized

flux as the ratio of the area of a bundle of rays in the planet planeρdρdθto the area of the bundle in the observer planeρ⁰dρ⁰dθ⁰, attenuated by the gas or scattering opacity,

Φ(ρ, θ) = ρdρdθ

ρ⁰dρ⁰dθ⁰exp[−τ_obs(ρ, θ)]. (4.11) To evaluate Eq. 4.11, we must first calculate the deflection angleΘ(ρ, θ), then trace the deflected rays to their destination (ρ⁰, θ⁰) in the observer plane. In terms of the components of Θin the ˆρ and ˆθdirections, this transformation can be expressed as

x⁰(ρ, θ) =ρsinθ+ Θ_ρsinθ+ Θ_θcosθ, (4.12)

y⁰(ρ, θ) =ρcosθ+ Θ_ρcosθ−Θ_θsinθ, (4.13) ρ⁰(ρ, θ) =

q

x⁰²+y⁰², (4.14)

θ⁰(ρ, θ) = tan⁻¹(x⁰/y⁰). (4.15) 4.4.4 Atmospheric model

In order to determine the deflection angle of a ray penetrating Titan’s atmosphere (Eq. 4.9), we need a model for the refractivity ν(r, φ) as a function of radius and co-latitude. The model must include the effects of planetary rotation and zonal wind, as well as the varying force of gravity over this broad region of Titan’s atmosphere. We can describe the rotation of the atmosphere as the sum of the rotation rate of Titanω_s and that of zonal winds with a speed Vw(r, φ),

ω(r, φ) =ω_s+V_w(r, φ)

rsinφ . (4.16)

In hydrostatic equilibrium, the pressure gradient will balance the sum of the gravitational and centrifugal accelerations experienced by a parcel of this atmosphere times the mass densityρ(r, φ) of the gas. We can express the r and φcomponents of this force balance as

µ∂P

∂r

¶

φ

=ρ^³−g+ω²rsin²φ^´, (4.17) µ1

r

∂P

∂φ

¶

r

=ρω²rsinφcosφ, (4.18)

whereg=GM/r² and the partial derivatives are taken at constantφand r, respectively.

For any realistic zonal winds, the gravitational acceleration will be much larger than the radial term of the centrifugal acceleration, so we can approximate Eq. 4.17 with the more familiar equation

µ∂P

∂r

¶

φ

=−ρg. (4.19)

Rearranging this expression,

1 ρ =−g

µ∂r

∂P

¶

φ

, (4.20)

we substitute it into Eq. 4.18,

−g µ∂r

∂P

¶

φ

µ∂P r∂φ

¶

r

=ω²rsinφcosφ, (4.21)

leading to the following partial differential equation which describes the shape of surfaces of constant pressure; _µ

∂r

∂φ

¶

P

=−1

gω²r²sinφcosφ. (4.22)

Equation 4.22 can be integrated along surfaces of constant pressure to determine their radius as a function of co-latitude. If these surfaces deviate only slightly from spherical, then we can mover² andgout of the integral, leading to the following approximate solution (Ingersoll, 1970)

r(P, φ)≈r(P, φ0) Ã

1−r(P, φ0)³ GM

Z _φ

φ0

ω(P, φ⁰)²sinφ⁰cosφ⁰dφ⁰

!

. (4.23)

An exact solution for r(P, φ) is given by Lindal et al. (1985), but Eq. 4.23 represents a sufficiently good approximation at Titan’s slow rotation rate. The integral in Eq. 4.23 describes the shape of surfaces of constant pressure, which for constant composition and temperature will coincide with surfaces of constant refractivity. In the context of this simplified model, Eq. 4.23 can be thought of as an altitude correction to the pressure structure of Titan’s atmosphere, due to the presence of zonal winds.

We compute the pressure as a function of radius at any fixed co-latitudeφby substituting the ideal gas law into Eq. 4.19 and expandingg,

1 P

dP

dr =− µ kT

GM

r² , (4.24)

where k is Boltzman’s constant and µ is the mean molecular weight. Solutions to this differential equation will be of the form

P(r, φ) =P₀exp

·µGM kT

µ1

r − 1

r(P0, φ)

¶¸

, (4.25)

where the pressure is normalized toP0 on a reference surfacer(P0, φ) whose shape is given by Eq. 4.23. The refractivity ν(r, φ) can be expressed in terms of the pressure P(r, φ) and the refractivity of a gas of identical composition at standard temperature and pressureν_STP as

ν(r, φ) = P(r, φ) kT

ν_STP

L , (4.26)

whereL is Loschmidt’s constant. Substituting Eq. 4.25 into Eq. 4.26, we can compute the refractivity at any location in the isothermal region of Titan’s stratosphere.