SPACECRAFT

42

SPACECRAFT

T the angle rp between T and Vsc may be calculated:

or, with X= cl/0 •

rp =arccos[_:!:_]

Vsc

rp =arccos [

;:!c ]

( 16)

(17)

With rp₆₁ known from the trajectory data and rp calculated from the Doppler data, the bending angle, 'f, may be calculated as follows:

(18) Once

'/1

is known, b, the ray asymptote, may be calculated:

b = rsc sin('/!+ ex) . (19)

Thus from the Doppler and trajectory data, b, the ray asymptote, and

1/t.

the bending angle can be determined as a function of time throughout the occultation. The index of refraction and radius at any given level in the atmo- sphere may be calculated using an Abel integral transform as follows (Kliore, 1972)

As described in Appendix 1. b (r_{0 )} = r₀n (r_{0 )}, thus b (r₀₎

To= n(ro)

(20)

(21)

44

The result is a profile of the index of refraction as a function of radius.

S. Procedure: Press~Temperature Profiles from Refractivity

Once the index of refraction of the atmosphere is known as a function of radius, atmospheric density. pressure and temperature may be calculated.

According to the Lorentz-Lorentz law for gases, the atmospheric density is pro- portional to the refractivity, N:

p(r) =

N(r)[#Q-]

⁽²²⁾

1-L is the molecular weight of the gaseous atmosphere (which can be taken to be C0₂ and N₂ for our purposes here).

(23)

where

f

co₂ and

f

N

2 are the volume mixing ratios of C0₂ and N_{2 •} respectively. and J.lco₂ and MNe are their respective molecular weights. R is the gas constant and

Q

=

0.269496(494.5 j co₂ + 294.4

fN

₂⁾ (24)

(Essen and Froome, 1951). Q describes the dependence of the refractivity on the atmospheric density and composition from laboratory measurements (the units of Q are K/mbar).

The pressure may be calculated assuming hydrostatic equilibrium:

where

r

p (r) =-

J

^{pgdr' +Po}

ro

Po= p(ro) -R To(ro) 1-L

(25)

(26)

where 1-L is the molecular weight of the atmosphere, R is the gas constant. and

T₀ is a guessed quantity at r. high in the atmosphere. Pressure profiles for difierent values of T0 converge at about 100mb. The temperature profile may then be calculated from the equation of state assuming the atmosphere is ideal:

T(r) = El!:.l_ 1!...

p(r) R (27)

The Venus atmosphere is approximately ideal at all altitudes above the critical refraction level.

4. Procedure: Refractive Defocussing Profiles from Bending Angle

The refractive defocussing equation used to compute the loss of power due to spreading of the beam is for a spherically symmetric atmosphere (Eshle- man et al., 1980). The derivation of the equation has been described in Chapter 11. The equation is

(28)

where R2 = vr}c-^b2 and R2o Tsc. ^b and R' are described in Figure 11c. In terms of decibels, the refractive defocussing, AREF· is

(29)

5. Errors

Temperature-pressure profiles are calculated from the refractivity profiles derived from the Doppler data as described in the previous section. In order to calculate the refractivity from the Doppler or bending angle data, an Abel integral transform must be used. Due to nonlinearity, errors inhC'rent are not directly calculable; however, the errors may be determined indirectly by recalculating the bending angle from a perturbed refractivity profile. These errors are a result of numerically approximating the Abel integral. Thus a

46

perturbation is a sudden change in the refractivity which introduces an error from the numerical approximation.

The refractivity is perturbed by a known amount, the associated bending angle is calculated from the perturbed refractivity profile and the resulting per- turbcd bending angle is processed as original data. The integral used in the reverse calculation is:

b (ro) ¹ :n dr

n r

(30)

In order to change this equation to an integrable form, a method developed by D.O. Muhleman (private communication) is used. A dummy vari- able, z, is used to simplify the integral as follows:

or

z =rn

dz

=

^ndr + r - - d r dn dr

dz

=

^dz

=

^dr

⁺

^{_L dn dr}

nr z r n dr

.L

^{dn dr}

n dr

dz dr

= - - -

_{z r}

(31) (32)

(33)

(34)

Substituting the l.h.s. of equation (34) into integral (30) and separating the resulting integral into two parts gives

I .

⁽³⁵⁾

The first integral on the r.h.s. of equation (35) is easily integrated:

J,r.

^{dz 1 b (ro)}

r

=

^cos-1

_~(;o)

b (r₀)

zvz

^{2 - b{ro)2 b (r0) z}

1 cos-^{1 b} (r_{0 )}

(36)

=

b (r_{0 )} r.

The second integral on the r.h.s. of equation (35) is just ~. the angle between r ₀ and r:

(37)

The integral equation (35) can be rewritten as:

r

^{1 -1 b{ro)}

1/J(r0) = -2b (r0)

l

^{b (ro) cos r.} b (ro) (38) or

r

^b(ro)

I

1/J(r0)

=

^-2lcos-1 _r. ^{- ~ (39)}

where b (r_{0 )} = n(r₀)r₀. This equation may also be derived geometrically from Figure 11b.

The only remaining obstacle to using equation (35) or {39) to calculate the bending angle is the integral ~ (equation {37)). A mathematical trick is necessary to evaluate the integral. The term (r²n^{2 -} b (r0)2

) is expanded in a Taylor series around r ₀:

But r²n ²

lr

₀

=

^{r6n (r}0 )2

=

^{b (r}0 )2 and the first term on the r.h.s. of equation ( 40) is zero. After differentiating the remaining equation,

48

+ higher order terms . (41)

For purposes of numerical calculations, c (r_{0 )} may be defined as

(42)

which simplifies equation (41) to

(43) near ro.

The mathematical trick is to add and subtract

1 (44)

as follows

r_ 1

r

¹

~=

^b(ro)J r₀ ^-r

l

r n ^{2 2} - ^b2( r₀ ⁾

1 dr

r

....JCf.T05 vr -

^{r 0 (45)}

Equation (45) may be divided into two parts -the integral from r₀ to

rti

(ncar r₀₎ and the integral from

rr1"

tor ... The r.h.s. of equation (45) is zero in the region r₀ to rtf because Vr²n² - b (r_{0 )}²

=

^{Vc (r}0)(r - r_{0 )}² near ro. ~then becomes

--;::::::;::::=~

_VC[TJ ¹

_{...;r -} '-;::::==-]

_To ^dT

+ b(To)Jr. l_ dT

r₀ T ~ (46)

Simplifying the second integral in equation (46) results in

follows:

~= b(T0)

J

r.

r•

0

2b(T0)

-~~-

-T 0

+ tan

..Jc (To)To To (47)

Summarizing, 1/I(T0) may be calculated from the refractive index, n (T) as

(48)

where '«r_{0 )} is given by equation (47) for any n(r).

Figure 12 shows the original bending angle and the bending angle calcu- lated from the refractive index using equation (48). There is essentially no dis- cernible difference between the two curves if no perturbations are introduced.

ln order to determine the efiect on the pressure-temperature profiles of a perturbation in the refractive index, the procedure outlined in Figure 13 is fol- lowed. The original refractive index profile, n₁ is perturbed as desired to pro- duce

n, .

The bending angle

1/lp

corresponding to this perturbed profile is calcu- lated using equation (48).

1/lp

is assumed to be the raw data. From

1/lp.

a new index of refraction is calculated, ^"npp· The difference between nP and ~P

represents the additional error to the refractive index introduced by the Abel transform. It is possible the transform may hide or accentuate perturbations in

50

Figure 12. The original bending angle and the bending angle calculated from the refractive index. There is essentially no discernible difference between the two profiles.

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E

- _~

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6092

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ORBIT 9N