Arrangement for the computation of mean intensities and fluxes in an atmosphere given on log τ-scale

1. Quonr.

Sprctrosc.

Radior.

Transfer.

Vol.

3. pp.

209-210.

PergamonPressLtd., 1963. Printedin GreatBritain

ARRANGEMENT FOR THE COMPUTATION OF MEAN INTENSITIES AND FLUXES IN AN ATMOSPHERE GIVEN ON LOG T-SCALE G. ELSTE* Cilittingen

IN THE computational scheme we use at Gijttingen for solar type atmospheres, the temperatures and pressures are given as functions of log TO(referring to h 5000 A)

log

Q=

10

5

j

logPe

1

IogP,

1

logKo/Pa

and the theoretical computed connection between the log T,-scales log

log

70

?4

Since the integrals for the mean intensity as a function of depth (UNDERHILL,equation ii) involve the first exponential integral having a logarithmic singularity at the zero point, we transfer to the log t-scale. &l TA 2 Mod L,{f(N

=

+*

j- f( 7h - #El(t) -U2

dlog t +

j-(~~ + t) t El(r) dlog r s --Q)

in which the function t El (t) behaves quite well as is shown in the figure. Both integrals can be evaluated numerically. The critical point in this scheme lies in the first of the two integrals. For large values of 7h the function.f( 7A- t) changes very rapidly from.f(~J to f(0). We hope to solve this problem soon. For the numerical integrations we normally use a generalization of the Gauss-Encke quadrature formula for constant stepwidth, in which one takes four points of the integrand for a second order interpolation, while the integral is taken between the two middle points.

s

“1 f(x)dx

= y(

(fo -fi) -([xi -xo12/6)

wz-fll/[X2 * Present address: Astronomy

Dept, University

-x11)

-u1

-Ibl/[x1

-xol,)/[x2-x04

of Michigan, Ann Arbor, Michigan. 209

G. ELSTE

210

log 1 E,(t) -0.5

/

- 14

,/’

/

r

‘\

! \

\

\

‘\

,‘/ -FE

\ I -2 ---P

log r

I

I

-1

0

i

FIG. 1

This generalization is needed, since for equal steps in log 70 the log TA-intervals are not necessarily of equal width.