The area of the visible disk, the area of the ansae (visible rings not overlapping the disk), and the area of overlap (visible rings overlapping the disk) were analytically determined to account for the thermal emission from the rings. These parameters were available in the ephemeris

Erickson, E. F.

Matthews, S.

English

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https://ntrs.nasa.gov/search.jsp?R=19770012033 2020-03-22T11:26:25+00:00Z
—r—
(NASA — TM—X-73204) ANALYTICAL CALCULATION OF	 N77-18977
T116 AREAS OF SATURN , S DISK AND RINGS (NASA)
7 p IIC A02/M A01	 CSCL 03A
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NASA TECHNICAL
	 NASA Tivi X73,204MEMORANDUM
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I-chmary 1977
I Repurt No ;	 Grnrernntent Accession No 3	 Recipients Catalog No
NASA TM X-73,204
• Title and SUbMIe 5	 Report Date
ANALYTICAL. CALCULATION OF THE AREAS OF SATURN'S 5 Pe,formuq orgonrnuon Code
DISK AND RINGS
0	 PerlormrngOtganrtatron Report No7 Authotlsl 
	 —!
S.	 Matthews and E.	 F.	 Erickson A-6906
10. Work Unit No
352-03-03-029	 Performing organization Nuns and Add r eu
NASA-Ames Research Center 11
	 CnntraKt or Giant No
Moffett	 Field,	 Calif.,	 94035
13	 Type of Hepurt and Period Covered
Technical Memorandum12	 Spunsor ng Agency Name and Address
:National Aeronautics and Space Administration to	 %ponsorrriq Agency Code
Washington,	 D.	 C.,	 20546
15 Supplementary Notes
16 Ahstratt
Determination of the thermal emission from the disk of Saturn at
wavelengths in the vicinity of the thermal peak (-50 um)
	
is complicated by
the fact that infrared telescopes currently operating at these wavelengths
cannot	 spatially separate the rings from the disk. 'lo account for the
emission from the rings,	 the area of the visible disk, 	 the area of the
ansae	 (visible rings not overlapping the disk), 	 and the area of overlap
(visible rings overlapping the disk) must be known. The calculation pre-
sented here describes the analytical determination of these areas from
parameters available in the ephemeris.
17 Key Words (Suggmted by Author(s)) 18	 Distribution Statement
Astronomy Unlimited
STAR Category - 89
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Unclassified Unclassified 6 $3.25
'For sale by the National T- . hrtcvl Information Serv.ee, Springfield. Virginia 22161
7-7- 1 1 r	 fI	 /
ANALYTICAL CALCULATION OF THE AREAS
OF SATURN'S DISK AND RINGS
S. Matthews and E. F. Erickson
Ames Research Center
SUKMARY
Determination of the thermal emission from the disk of Saturn at wave-
lengths in the vicinity of the thermal peak (-50 um) is complicated by the
fact that infrared telescopes currently operating at these wavelengths cannot
spatially separate the rings from the disk. To account for the emission from
the rings, the area of the visible disk, the area of the ansae (visible rings
not overlapping the disk), and the area of overlap (visible rings overlapping
the disk) must be known. The calculation presented here describes the
analytical determination of these areas from parameters available in the
ephemeris.
Saturn's rings and disk make comparable contributions to the thermally
emitted flux from the entire system. If one assumes that the optical depth of
the rings is not necessarily high, and that the only significant ring effects
are due to the A and B rings, then five areas are important in estimating
the relative contributions of the rings and disk. These are:
Qvd = unobscured (visible) area of the disk
QA ,Q B = visible area of the A and B rings respectively
wA ,c.,B = area of disk obscured by the A and B rings respectively
The brightness of the entire system is described by brightness tempera-
ture T, which is related to the rings' temperatures, TA and T B , and the disk
temperature, Td , by
Bv(T) 2t - Bv(Td)(Ovd + e-'AwA + e-TBwB)
+Bv(TA) (1 - e TA )f2A + Bv(T ii ) (1 - e TB)QB
Here Rv is the Planck function,
	
Q t
 = S?vd + 'QA +Q B	(1)
Is the total system area, and T A and -i B are the optical depths of the rings.
130th the disk and rings are elliptic-1 in appearance to the observer. In
rectangultr coordinates the equation of an illipse is
a
7 y2
+ b2	 1	 a > b	 (Z)
If we transform this to polar coordinates we have x = r cos e, y - r sin A,
SO
r2 C08 2 9 + sfr 2 e) - 1 or r(0) - ab(a 2 sin g 6 + b 2 cos t 9) -1/2	 (3)
l az	b2 1
The area of an ellipse is crab. our notation for the Saturn geometry is shown
in figure 1. Thus, for the aroblem at hand, we have
Qvd - naobo
 - wA
 - wB 	(4)
QA = ir(a l b l - a 2 b 2 ) - wA
(5)
"B
 - n(a3b3 - a 4 h 4 ) - w8 )
it we note that (see ref. 1)
a,	 b2
al = bl - 0.8801
	
(6)
a 3
	b3 - 0.8599	 (7)
a 1 	hl
and
a4 = b4 - 0.6650	 (8)
a 1
	t l
then the quantities 32, b 2 , a3, b 3 , a 4 , and b 4 can he eliminated in (5) so
that
QA = 0.2254nalbl - WA
(9)
QB = 0.2972nalbl - wB
ao , bo , a l , and b l can be obtained from the ephemeris.
There remains simply to calculate wA and w R . To determine the angles
e i , i = 1. to 4, we note that
r i ( e i ) = rc (e i ) or r1 2 (ei) - ro 2 ( e i) 	 (10)
Then
2
22a i b i ao2bo2
ail sin'
	
6 1 + bi2 cos t 0 i ao 2 sine 0 i + bog cos t 0i
which can be solved for	 tan 01
2
1/?
t i = tan Ai
^,^ 2aoai	 _
b° ; for	 bi <_ bo (12)	 r,.
If	 b i should be greater than	 bo , t i (01	 (71/2)J.	 Now
wA = 2fe0^ fr r° r dr d0 + 21 /2frzl	 r dr d0 (13)
= 1h21 ( ro t - r2 2 )d0 + f0 tt
/ 2	 ( r 1 2 -	 r 2 2 )d0 (14)
10 2 1 ro t d0 + f01/2 r 1 2 d0 - f0 Z' 2 r2 2 d0 (15)
= a obo
a	 01
tan-1 (bo tan 0))	 + a 1 b 1	 tan
a
bi 	 tan
n/2
0))
0	 02 /	 el
-a 2 b 2 tan- 1 ( a2 tan 0	
n/2
)
/
2
(16)
02
since
fr 2 d0 - a2b2
	
d0
= ab	 tan-1 
^a tan	 0)b (17)a2 sin 2 0 + b 2 cos t	0
Rewriting (16), and from a similar expression for w B , we find
wA = aob o
Itan
-11 bot11- tan-11 bot2)]+ a1b1[2-Lan-(1tl)]
 \ o 	/	 ` o	 1
r
	a
-a 2 b
21
2 - tan
-1 (b2
 t2)]
	
(18)
2
3
and
wB	 aoboltan-llbo t3) - tan-1
\bo t4 /J + a3h3L2 - tan
-1 ^b3 t3,
J0	 o	
-a,,b 4 2 - tan- l(b4 t4)J
	
(19)
1L	 4
From (6), (7), and (8) we obtain finally
wA 0 aoboltan- l(bo t l) -tan -f l abbo t 22 I
 
+albll0.1127r- tan- l(bl
 ti)
+0.7746 tan-l (! t2)
J 	
(20)
1
and
wg - aobo l tan-I ^!o t3) -tan - '(b-O0 t 4 1
J 
+a l bs I0.1486r -0.7394 tan -l (b t3)
a	 `
+0.4422 tan -1 (b l t 4 I	 (21)
1
As an example, we calculate the areas for January 27, 1976. On this date
the ephemeris parameters (in arc seconds) are
ao - 10.295	 bo - 9.215
a l - 23.19
	
bl - 8.49
Then, from (1), (4), (9), (20), and (21) one obtains in square arc seconds the
areas
Qvd -	 259.02
:2A -	 127.91
Q B -	 156.25
WA -	 11.52
wg -	 27.49
,Z t -	 543.18
REFERENCE
The American Ephemeris and Nautical Almanac, issued by the Nautical Almanac
Office, United States Naval Observatory. U. S. Government Printing
Office, Washington, 1976, p. 416.
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Analytical calculation of the areas of Saturn's disk and rings

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