LDEF temperature histories: a simple theory

•

October 1990

2 APR.

f991

LDEF

Tem perature Histories

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26

A Simple Theory

-

29 H DELFT

P. C. Hughes

UTIAS Report No. 340

..

.,

Submitted September 1990

•

October 1990

LDEF

Temperature Histories

A Simple Theory

P. C. Hughes

© Institute for Aerospace Studies 1990

UTIAS Report No. 340

eN ISSN 0082-5255

•

Table of Contents

Foreword

Summary

1

Introduction

1.1

Semi-Quantitative Features of the LOEF Temperature Oata

1.2 Physical Considerations Included in the Theory

2

System Geometry

2.1

Inertial Reference Frame at Sun,

:Fa . . . .

2.2

Inertial Reference Frame at Earth,

:Fb . . . .

vii

ix

1

1

4

13

· . . . . .. 13

· . . . . .. 15

2.3

Equatorial Reference Frame for Earth,:F

e • • • • • • • • • • • • • • • • ,

15

2.4

Reference Frame Associated with LOEF Orbit,

:F

p • • • • • • • • • • • • • • •

17

2.5

Reference Frame Associated with LOEF Vehicle,:F

o • • • • • • • • • • • • •

19

2.6

Unit Vector

2

Associated with an Experimental Test Patch . . . .. 22

3

Time Dependence of Geometrical Factors

25

3.1

Time Dependence of

'TJe(t) ••••

3.2 Time Dependence of

Cab . . . . .

3.3

Time Dependence of

Oe

and

ie . . . . .

3.4 Time Dependence of 0 and

i . . . .

.

. . .

3.5

Time Dependence of

'TJ

• • • • • • • • • • .

3.6

Time Dependence of

À .

4

Conditions for Shadow

. . . . 25

. . . . 26

. . . . 26

· . . . 26

27

. . . . 28

4.1

The Geometrical Condition for Earth-Shadowing (Eclipse)

30

30

4.2 The Geometrical Condition for Self-Shadowing. . . .

.

. . . ..

32

III

5

Normalized Solar Heat Flux to Patch

35

5.1

Orbital Motion

Only

. . . 35

5.2 Orbital Motion, Plus Precession . .

.

. .

. . .

. . . . .

.

. . . .. 37

5.3

Earth-Shadowing vs. Self-Shadowing . .

.

. . .

.

.

.

. . . .. 40

5.3.1

Self-Shadowing Only . . .

~

. .

.

.

.

. . . . .

.

. . . .. 40

5.3.2

Earth-Shadowing Only

.

.

.

. . . .

. .

.

.

. . . .

.

. . . .

.

. .. 45

5.3.3

Self-Shadowing and Earth-Shadowing Combined . . . .. 48

5.4 Orbital Motion, Plus Precession, Plus Annual Variation . .

.

. . .

.

. .. 48

5.5

Effects of 16-Hour Sampling

. . . .

. . .

.

.

.

.

.

. . . .. 49

6

A Simple Thermal Model

51

6.1

Heat Flow to the Vehicle

. . .

51

6.2 Temperature of the Vehicle,

Tv

.

. .

.

. . .

. .

.

.

. . .

.

. .

.

.

.

. . ..

52

6.3 Heat Flow to the Patch, Directly from Sun . . . .

.

.

.

. .

.

. . . .. 53

6.4 Heat Flow

to

the Patch, from LDEV

. .

. . . .

.

.

.

.

.

. .

.

. . . .. 54

6.5 Temperature of the Patch,

T .

.

. . . .

.

. .

.

.

. .

.

.

. . . .

.

.

.

. ..

54

6.6 Values of the Thermal Parameters. .

. .

. . .

.

. . .

.

. . . .

.

. .. 55

7

Temperature Histories

56

7.1

Vehicle Temperatures

.

.

.

.

. .

. . .

. . .

.

.

.

.

.

.

.

. . . .. 56

7.2 Test Patch Temperatures .

. .

. . .

.

. .

.

.

.

. . . . .

.

. . .

.

. . . .. 61

8

Concluding Remarks

65

9

References

66

A

Computer Code

67

List of Figures

1

LDEF -

The Long-Duration Exposure Facility.

2

2

The set of UTIAS test patches.

.

. . . .

3

3

LDEF temperature data for tbermal gauge #1.

6

4

LDEF tempera

'

ture data for tbermal gauge #2.

7

5

LDEF temperature data for tbermal gauge #3.

8

6

LDEF temperature data for tbermal gauge #4.

9

7

LDEF temperature data for thermal gauge #5.

10

8

LDEF temperature data for tbermal gauge #6.

11

9

The Sun-centered reference frame, Fa, and tbe (parallel) Eartb-centered

ref-erence frame,

Fb.

. .

. . .

.

.

.

.

.

. . . .

.

.

.

. . . ..

. . . . ..

14

10

Tbe geo-equatorial reference frame, Fe. . . . .

.

. . . ..

16

11

Tbe reference frame associated witb LDEF's orbital plane, Fn. . . .

.

. . ..

18

12

The reference frame associated witb LDEF's orbital plane and orbital perigee,

F

p • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

20

13

Tbe reference frame associated witb LDEF in orbit - tbe "orbiting frame"

- F

o

. .

.

. .

.

. .

. . . .

. .

.

. . .

.

. . . .

21

14

Tbe unit normal vector outward from tbe test pateh,

~.

. . . .

23

15

Tbe geometrical condition for sbadowing from Earth (eclipse).

31

16

Tbe geometrical condition for self-shadowing. . . .

33

17

Normalized solar flux, over one year, witb no LDEF orbital precession, and

witb no rotation of Eartb around Sun.

.

. . . .. 36

18

Normalized solar flux -

over tbe first two days only - witb no LDEF orbital

precession and witb no rotation of Eartb around Sun. . . .. 37

19

Normalized solar flux, over one year, witb LDEF orbital precession, but witb

no rotation of Eartb around Sun. .

.

. . . . . 38

20

Normalized solar flux -

over tbe first two days only -

with LDEF orbital

precession, but witb no rotation of Eartb around Sun. . . .. 39

21

Normalized solar flux, over one year, witb LDEF orbital precession, witb no

rota ti on of Earth around Sun. Self-sbadowing only. . . . 41

22

Normalized sol ar flux -

from Day 0 to Day 2 -

witb LDEF orbital

preces-sion, but witb no rotation of Eartb around Sun. Self-sbadowing only. . . 42

23

Normalized solar flux -

from Day 10 to Day 12 -

witb LDEF orbital

pre-cession, but witb no rotation of Eartb around Sun. Self-shadowing only.. 43

24

Normalized solar flux -

from Day 0 to Day 0.2 -

witb LDEF orbital

pre-cession, but witb no rotation of Eartb around Sun. Self-shadowing only.

44

25

Normalized solar flux, over one year, with LDEF orbital precession, with no

rotation of Earth around Sun. Earth-shadowing only. . . .. 46

26

Normalized solar flux

-

from Day 0 to Day 2 -

with LDEF orbital

preces-sion, but with no rotation of Earth around Sun. Earth-shadowing only. . 47

27

Normalized solar flux -

from Day 0 to Day 0.2 -

with LDEF orbital

pre-cession, but with no rotation of Earth around Sun. Earth-shadowing only. 47

28

Normalized solar flux -

from Day 0 to Day 0.2 -

with LDEF orbital

pre-cession, but with

no rotation of Earth around Sun. . . .. 48

29

Normalized solar flux, over one year, with LDEF orbital precession, and with

Earth rotation around Sun.

.

.

. .

.

. .

. . . .

50

30

Vehicle temperature history ror the first 2 hours and 20 minutes. . . .

57

31

Vehicle temperature history ror the one week. . .

.

. . . .

58

32

Vehicle temperature history ror one week, as sampled every 16 hours.

59

33

Vehicle temperature history ror one year, with 16-hour sampling. . .

60

34

Test patch temperature history ror one year.

. . .

62

35

Test patch temperature history for one year, with 16-hour sampling.

(Com-pare with Figure 4.) . . . .

.

. . . . .

.

. .

.

. . . .

.

. . . .. 63

36

Temperature history ror one year, with 16-hour sampling, ror a different test

patch. (Compare with Figure 3.) .

. .

. . . .

.

. . . .

..

64

Foreword

The author wishes to thank the tol/owing individuals:

• Dr Rod Tennyson -

for bringing this prob/em to the author's

attention, and for he/pfu/ discussions;

• Dr Don Morison -

for providing data, and for he/pful discussions;

• Vince Pugliese -

for checking, using the 'Mathematica' computer

code, the inner products used in eva/uating the "shadowing

conditions" of

§4;

• Simon Grocott -

for preparing the computer plots in §5 of this report.

Summary

The temperature of

a

particular area (an experimental"test patch'? on the

surf ace of LOEF varies with time in

a

manner that depends on

a

large

number of factors. The direct input from Sun depends on whether LOEF

is in Sunlight; if in Sunlight, on whether the test patch can actually see

Sun; and if it can see Sun, on the angle of incidence of Sun's radiation.

The heat input to the test patch from the rest of the LOEF vehicle, also

important, depends on the average temperature of the LOEF vehicle, which

in turn depends on its eclipse history. This report examines all these factors

and uses

a "

multiple-time-scale" description (corresponding physically to

"orbital, " "precessional" and "annual" time scales) to build an understanding

of the chief features observed in the f1ight data. Geometrical, orbital, and

thermal models are developed. Numerical calculations based on these

models are shown to be in qualitative agreement (and usually even in

semi-quantitative agreement) with f1ight data. Thus all the salient features

of the in-orbit temperature data can be explained in terms of the physical

phenomena descibed.

•

1

Introduction

Through technical conversation with a colleague (Dr R C Tennyson), the author became

aware of certain temperature data collected in orbit on a set of experimental test patches on

LDEF -

the Long Duration Exposure Facility. The LDEF satellite is shown

in

Figure 1 and

the test patches under consideration are shown in Figure 2.

The primary purpose of LDEF was not to collect temperature data. Nor was the primary

purpose, even of the test patches, to collect temperature data; it was, instead, to evaluate

experimentally the effects of the LED space environment on a variety of material specimens,

as explained by

TENNYSON

et al. [1990]. Nevertheless, temperature data were collected as

a means of evaluating the space environment to which the test specimens were subjected and

it

is therefore a helpful ancillary objective that the nature of the temperature data be clearly

understood.

This report addresses the question of how to explicate the qualitative characteristics of

these temperature data in terms of simple physical principles. Thus, it is not the goal herein

to obtain "exact" agreement between the math models derived below and the LDEF data, but

instead to use these models as a tooI to gain physical understanding.

1 .. 1

Semi-Quantitative Features of the LDEF Temperature Data

The temperature data shown in Figures 3-8 refer to the several material samples on the test

patches, as measured over a period of about one year. For present purposes the qualitative

features of these data can be described in terms of

th ree time scales, as follows:

I.

Fast-Time-Scale Changes: These are the relatively rapid oscillations (of the

order of two or three days).

11.

Slow-Time-Scale Changes: These are the relatively slow oscillations (of the

order of 45 days).

111.

Very-Slow-Time-Scale Changes: These are the even slower oscillations (of

the order of 300 days).

Introduction

2

11

•

Figure 1:

LDEF --

The

Long-Duration Exposure Facility.

Introduction

3

Figure 2:

The

set

of UTIAS

test patches.

Introduction

It is the purpose of this report to demonstrate the physical origin of these three superposed

oscillations, and to discuss several collateral issues as weIl.

The discussion in this report is intended to be arithmetically semi-quantitative and

phys-ically

motivated. Not wanted are elaborate systems of partial differential equations, nor are

complex software packages for orbit dynamics and therm al analysis. Rigorously quantitative

comparisons between the thermal data and analysis are left to others who may have the

re-sources

and the interest to make such comparisons. The object here is to contrive a simple

physically-transparent theory that can explain the physical origins of the three time scales

mentioned above by demonstrating th at the essential characteristics of the data in Figures 3-8

can be reproduced and explained using that theory.

One last important note on the data shown in Figures 3-8.

The circular dots are the data.

The lines joining the dots have simply been drawn by the figure-maker to "join the dots."

Thus,

the solid fines do not represent data. As we shall see in this report, the solid lines do

not follow -

even approximately -

actual (instantaneous) temperature histories, except in a

time-averaged (slow-time-scale) sen se.

1.2

Physical Considerations Included in the Theory

The following physical/geometrical effects will be included in the "simple theory" developed

in this report:

1. Where is the test patch on WEF?

2. Where is LDEF in its orbit about Earth?

3. What is the (precessed) orientation of LDEF's orbit with respect to an

iner-tially fixed geo-equatorial plane?

4. Where is Earth in its orbit about Sun?

5. When is LDEF shadowed by Earth? -

That is, when is LDEF in eclipse?

6. When is there "self-shadowing1" -

That is, when is there shadowing of the

test patch by WEF itself!

7. What are the most basic effects ofthermaltime lags? - That is, what are the

consequences ofthefact that the patch temperature is not simply proportional

U T I A

S

, . - - - -

-'

..

'

.

Introduction

to the instantaneous radiation (energy) flux from Sun?

8. Bearing in mind that the temperature is sampled only every

16 hours (I),

wOOt are the effe cts of this sampling?

At the same time, the following simplifications will

he

made:

9. WEF's orbit around Earth is assumed always circular.

10. The inclination of LDEF's orbit with respect to Earth's equatorial plane is

assumed constant.

11. Attitude librations are neglected; their effects,

ij any, are assumed

10

average

out in a relatively short time span.

A number of other specific assumptions will also

he

made; however, these are sufficiently

technical that the best place to introduce them is where they occur in the theory.

U T I A

S

Introduction

6

..

200

Thennal Gauge #1

ST AINLESS STEEL -

CALIBRATION TUBE

150

{~

l

~

~

.

.M!

I

n~

~

.1

jll~

-

r

~I

LL

₁

I

,

ï

I

1

I~

0

~

I

i~~

J

100

1

1

Q) I ~ ~

j

l

1\

::J

-

TI

i

1

I

cu

1

i

1

~ Q)

.

l

.

.

I' I

a.

i

•

_I I i •

I

~

J

E

I I I . !

f

.

_i

_.

₁

• I

I

r

i

l

l

II

Q)

50

I

l

1

.

\

_I

_{I !}

l-J

I I

·

,

;

,

,

,

I

~

~i"./V'\lV~

.

_·

,

}

i

_~

i

·

\

I

J

~

·

\

j

~

,

~

~

:

\'...'.Ï'

"i'

~

,

0

o

100

200

300

400

Elapsed Time

trom

Initiate (days)

Figure 3:

LDEF temperature data for thermal gauge #1,

Introduction

7

200

Thermal Gauge #2

11

GRAPHITE/EPOXY • 934/T300

150

:

!

I1

11

11

1

I1 '}

u:-

₀ Q)

100

' -:::J

-

('Ij ' -Q) Q.

E

50

Cl)

f-0;

i

'

,

:

1

1II

~

.

V

I

,

I

1

.1

Ii!

'L

₁₁

_'

0

{\l

j

~~

rt

~\~~

.

~

f

~~~

l

~ru

ll

'

~

~k1'

I~

-50

0

100

200

300

400

Elapsed

Time

trom Initiate (days)

Figure

4:

LDEF temperature data for thermal gauge #2.

Introduction

8

200

Thennal Gauge #3

KEVLAR/EPOXY

•

Sp·32S

150

:

:

;

I

-~

.

r

_l

_i

_l

~1

u..

_I

~

1

0 al

100

I

...

~

::J

-

~

...

al

a.

_I

E

I

_\

~

Cl)

50

II

I-l

~

_i

~

'\

i

i

rt

_~

~

i

i

_.J

,

I

1t

,f

~ ;

.

..

,

1

1

_t

o

.

_"t

I'

~

;.;,/

»

.'}j;

-5

0

'---'----'--_L.._~--'----'-_.L..! ~--, --'-_-L---1---'-_-L---L...---l_-L---L...---1---l

o

100

200

300

400

Elapsed Time trom Initiate (days)

Figure 5:

LDEF temperature data for thermal gauge #3.

Introduction

200

150

CL

_o

100

Q.) '-::J

-

cu

' -Q.)

a...

~

50

~

o

_.

f

!

₁₁

Ili

,

.

i

r,'

'I I1

·1

~

_.

•

Thennal Gauge #4

GRAPHITE/EPOXY • SP-288/T300

1

i

-50

~ï~

__

~~

__

L-~~

_ _

~-L

.

~

_ _

~-L

_ _

L-~~

_ _

L--L~

_ _

~-L~

o

100

200

300

400

Elapsed Time trom Initiate (days)

Figure 6:

LDEF

temperature data for thermal gauge

#4.

U

T I A S

Introduction

10

200

_{Thermal Gauge #5}

GRAPHITEIEPOXY • 520B/T300

150

I'

·

I

1

1

_ti

_T

'\'

_·1

1·

~I

T~

i

l

!

\\

'

l

i

ij

-

'

~i

l

_I

r

I L.L I 0

100

I

r

I.~

Q.)

1 \

I

:

....

::l

\

1.

1

1

...

cu

....

Q.)

0-E

1 \

~II

J

Q.)

50

1

\

\

\

\

1

1

I-\\1

!

\i

ti

J

_\

₁

₁

_a

₁

_{\ \}

Il

'

i

l

₁

~~

~

1

1

1

_!i!

.,t • • ;

I

j

;

-11

l

/

~~J\~~r~~~

~~

•

1;

_{.... j .}

I:

!

~t ~

_1\

₁

_\

,

1\ 1 ~. \ }\ I \

:

~

..

~'

r

.50

L-~

~~~~~-~~~~-

~~~~~~~~~~~~~~

o

100

200

300

400

Elapsed

Time from Initiate (days)

Figure

7:

LDEF temperature data for thermal gauge #5.

Introduction

11

200

Thermal Gauge #6

BORON/EPOXY • SP-290

150

.~

CL

o

100

Q) ~ ::J

-

co

~ Q)

a.

E

Q)

50

I-o

o

100

200

300

400

Elapsed Time from Initiate (days)

Figure 8:

LDEF temperature data ror thermal gauge #6.

2

System Geometry

In this section, we consider the purely geometrical problem of the angle of incidence

(or inclination) of a plan ar patch of material on the outside of LDEF with respect to Sun.

Quite a number of angles are involved here -

at least

seven,

as we shall see -

and hence

the calculation requires some careful technique. Some of these angles are constant; some are

time varying. These time variations will be considered in §3.

2. 1

Inertial Reference Frame at Sun,

Fa

We proceed by defining a series of reference frames, proceeding from Sun, in a sequence

of stages, to the patch on LDEF. The first frame, denoted

Fa,

is shown in Figure 9. The

origin

Oa

is located at Sun's center, and the frame is inertial (to the extent hat Sun's center is

not accelerating - an excellent assumtion for present purposes). The unit vectors associated

with

F

a

are denoted (using the notation explained by

HUGHES

[1986]) by

_....

al>

_....

a2' a3'

_....

and the

associated

vectrix

is denoted by

Fa,

_....

defined as

(1)

The unit

so/ar vector,

a vector from Sun aimed at the LDEF experiment -

aimed, to all

intents and purposes, at the center of Earth -

will be denoted

i.

As expl

a

ined in Figure 9,

A L1 A A ' _ T . [ COS

"I

e

]

~

=

~l

COS

"I

e

+

~2

S1l117e

=:;:

a

Sl~17

e

[

]

T

COS

"Ie

Si~TJ

e

:;:a

(2)

The time dependence of

11

e

(t)

will be considered in §3.1.

System Geometry

(Ecliptic plane)

Earth

11

(t)

e

Sun

Figure 9:

The Sun-centered reference frame,

:Fa' and the (parallel) Earth-centered

ref-erenee frame,

:Fb-U

T

I A

S

System Geometry

2.2

Inertial Reference Frame at Earth,

Fb

Still using Figure 9, we have the second reference frame,

Fb'

The origin of

Fb

is displaced

from

the origin of

Fa

by the di stance between Earth's and Sun's centers. No new angles are

thus introduced, however. In fact, The rotation matrix hetween

Fb

and

Fa

is just the unit

matrix:

(3)

and,

of course,

Cab

= Cra,

which in this case is just 1.

Note that, although

Ob

is at Earth's center, the frame

Fb

is not fixed in Earth, nor is

it

even an equatorial frame. The rotation of Earth on its axis is, for present purposes, irrelevant.

A geographic frame for Earth is, however, very important, and will now he considered.

2.3

Equatorial Reference Frame for Earth,

Fe

As

shown

in Figure 10, we introduce now an equatorial frame

Fe,

with the following

proper-ties: O

e

is located at Earth's center, and

_~

ê

3

coincides with Earth's north pole. It follows that

the

~C~2

plane is Earth's equatorial plane.

Based on Figure 10, we can show, using the notation of

HUGIIES

[1986], that

C

eb ~

Fe'Fr

~

...

C1(ie)C3

(Oe)

[~

0

_{o ][}

_cosn,

sin Oe

~l

COSZ e

sin i.e

- sin Oe

cos Oe

- SIn Ze

COS

Ze

0

0

[ COS

n,

sin Oe

sit~

i,

1

- cos

ie

sin Oe

cos ie cos Oe

(4)

sin

ie

sin Oe

- sin ie cos Oe cos Ze

where

ie

is the inclination of Earth's polar axis to the ecliptic plane ahout an azimuthal

direction given by O

e

. Over short time periods (like a few years) both Oe and ie are essentially

constant.

Note, however, that although Oe is at Earth's center, and although ê

_...

₃

coincides with

U T I A

S

System Geometry

15

"-.P---"""---I..-~·k2

Figure 10:

The geo-equatorial reference frame,

:Feo

System Geometry

Earth's north pole, the frame

Fe

is not fixed in a rotating Earth. (For one thing, as stated

previously, the fact that Earth is rotating is irrelevant for

OUT

present purposes.)

2.4 Reference Frame Associated with LDEF Orbit,

F

p

Thus far, we have located Earth's poles and Earth's equatorial plane with respect to the

ec1iptic plane. No particular orbit has been specified -

much less a particular direction on a

particular spacecraft in such a particular orbit.

In ge neral , any Earth orbit also lies (as a zero th approximation) in a fixed plane. As

shown in Figure 11, we can define this orbital plane, with respect to

Fe,

by its norm al vector,

using two angles,

i

and O. The

orbital plane frame

Fn -

specifically, the unit vectors

11,1

_...

and

~2

-

define the orbital plane. Thus,

Cne

~

Fn ':F;

_...

_...

C}(i)C

3

(O)

[~

0

_{o ][}

_cosn

sin n

~l

cos

z

sin

~

- sin

0

cos

0

- sm z

COSt

0

0

[

cosn

sinn

si~

i

1

- cos

i

sin

n

cosi cos

0

(5)

sin

i

sin

n

- sin

i

cos

0

cosz

And, of course,

een

=

C~e.

The parameter (orbital element)

i

is called the

orbital inclination,

and

n

will here be called the

argument of the ascending node.

In order for each orbit to have a unique set of orbital elements, these angles are limited

to the following ranges:

o

:s;

i

~

7 r ,

0

:s;

n

<

211"

(6)

Furthermore, the orbit normal,î is given by

_...

[

sin i sin

0 ]

=~;

-

sin i

C?S

0

cos z

(7)

U TI A

S

16

System Geometry

/

\

/

/

/

/

/

/

/

/

/

/

/

/

/

/

/

\

~I

\

/

/

\

/

\

/

\

/

\

/

\

/ /

\

/

\

/ /

\/

./

,.

'- orbital plane

Figure 11:

The reference f..-ame associated with LDEF's orbital plane,

:Fno

U T I A

S

System Geometry

The other "direction of interest," denoted ?'il, is also very important:

_....

(8)

We can now concentrate on the motion within the orbital plane.

As shown in Figure 12, we define the

perigee frame F

p ,

and in particular let the perigee

be located by a direction at angle

w

to

111

_....

.

within the orbital plane. Clearly

w

gives the

orientation

of the orbit within the orbital plane. In fact,

[

cosw

-s~nw

(9)

However,

we shall herein assume that

mEP

is always in an essentially circular orbit.

Thus the orbital eccenticity

e

=

0, and

w

is undefined. Therefore, without any essential 10ss

in generality,

Cpn

=

Cnp

=

1, and

Cpe

=

Cpn

ene

=

ene

Cep

=

C;e

=

e~e

(10)

(11)

with

Cne

being inferred from

(5).

The possible time dependence of

n

and

i

will be considered

in §3.4.

2.5

Reference Frame Associated with LDEF Vehicle,

Fo

Next, we consider the position of the spacecraft

(LDEF,

in Dur case) in the specified orbit.

Consider Figure 13, from which it may be deduced that the reference frame

Fa,

which is

fixed to the

nominal

spacecraft orientation in orbit, can be found from

F

p

as follows:

C2(1J)e2(~ )el(~)

C2(~

+

1J)el(~)

U T I A

S

System Geometry

1\

m

!

orbital

plane

Figure 12: The reference frame associated with LDEF's orbital plane and orbital perigee,

F

p •

U T I A

S

System Geometry

LDEF Orbit

Earth

_

.

_---_.

Figure

13: The reference frame associated with LDEF in orbit - the "orbiting frame"

-

F

o

-[COS

(t+',l

0

- sin

~~

+ "l]

[~

0

si~ ~

]

1

cos

2!:

sin

(~

+

7])

cos

(~

+

17)

0

2

0

• 11"

cos

2!:

- sm

2'

₂

[ -

s~n~

0

-C;Sq]

[~

0

!]

1

0

cos

7]

0

-

Slll1]

0

-1

[

-sin~

cos

1]

!]

=

0

0

(12)

cos

7]

sm

1]

The time dependenee of the orbital "anomaly" 7](t)

is the most rapidly changing variabie, as

will be discussed in §3.5.

U T I A

S

System Geometry

2.6

Unit Vector 0 Associated with

an

Experimental Test Patch

-+

FinaIly, having established a nominal frame associated with the LDEF vehicle in orbit, we

now move on to consider the orientation of the test patch on the vehicle. However, it should

he carefuIly noted that we are at this point assuming

1'10

attitude librations of the vehicle

with repeet to the nominal frame,

Fo.

Equivalently, if there

are

such lihrations, they will be

assumed negligible. Or, again equivalently, even if there

are

such librations, and even if they

are

not instantaneously negligible,

they will he assumed not large and to average out over a

time period smaller than the "fast"-time-scale changes identified in §1.1.

That is,

"roIl,"

pitch," and

"yaw"

will be assumed either identically zero, or negligihle, or

to be small and so rapid as to have no effect on the phenomena being studied. (If this category

of assumptions is not supportahle, an additional, body-fixed reference frame -

involving the

three angles of roIl, pitch, and yaw -

would he introduced at this point.)

As shown in Figure 14, we assume that LDEF has cylindrical geometry and that the test

specimen is a small, essentially planar article on the side surface. This being the case,

it

is

possihle to specify the normal to the experimental article by a single vector, denoted 0, which

_....

is normal to the vertical (the yaw axis, under present assumptions). lts location in the

_...

Ol-~

plane is specified by the angle

À:

[

. \ 1

T

SIn A

cO~À ~o

[

sin

À

1

o

=

01

sin

À

+

~

cos

À

=:F~

cos

À

~... -. -+

o

(13)

Since the experiment is at a fixed position on the spacecraft, the angle

À

is constant.

U T I

A S

System Geometry

22

---

--.---I

-

--

-

1

-...

- - [

·

- - - - . . , . , - : 0

-

-

/

o

- )

LDEF

Figure 14:

The unit normal vector outward from the test patch,

_....

iJ •

3

Time Dependence of Geometrical Factors

In this section we examine the time dependence of the geometrical factors introduced

inthe last section, and derive the analytical conditions for Earth-shadowing (eclipse) and

self-shadowing.

3.1

Time Dependenee of

TJe(t)

Seven angles were introduced in §2, namely,

We now consider the time dependence of these angles, beginning with "Ie.

The angle "Ie(t) is the "true anomaly" associated with the Earth's orbit about Sun.

There-fore, its time dependence is weIl known from Keplerian orbit theory.

If

t

_pe

is the peritime of

Earth's orbit about Sun, and 2a

e

is the major diameter of that orbit, and ee is the eccentricity

of that orbit, and Ps is the gravitational constant for Sun, then the algorithm for "Ie(t) proceeds

as foIlows:

1. Given

t.

where

That is, solve

ee

=

0.0167272

Ps

=

1.325

X

10

20

m

3

jsec

2

a

e

=

1.495

x 10

11

m

23

(1)

(2)

(3)

(4)

(5)

\

Time Dependence of Geometrical Factors

3. Then

71

e

(t)

is given by

7le

I§;+

ee

Ee

tan-

=

t a n

-2

1 -

ee

2

(6)

with

7l

e

and

E

e

being in the same quadrant.

If we can approximate Earth's orbit about Sun as being circular, then

ee

=

0, and

(7)

Note that when

(t - t

pe )

=

3.156

X

10

7

sec (i.e., one year),

7le

=

6.283 (i.e., 211").

3.2

Time Dependenee of

Cab

There are no angles involved in the "transformation" from

Fa

to

Fb,

as is evident from (2.3).

Therefore there are no time dependences to consider in this transformation.

3.3

Time Dependenee of

Oe

and

ie

The north pole of Earth can

he

assumed inertially fixed for present purposes. (In reality, there

is a slight periodic nutation of

i

e

with a period of 18.6 years, and a secular precession of

ne

with a cycle of 26,000 years.)

Thus, for our purposes,

n

e

=

constant

i

e

=

constant

We shall assume

n

e

=

0 and

i

e

=

23.5°.

3.4

Time Dependenee of

0

and

i

(8)

(9)

At the low altitudes of interest here, we can assume roughly that the LDEF orbit normal

precesses about Earth's north pole according to the formula

(10)

U TI A

S

Time Dependence of Geometrical Factors

where

re is Earth's equatorial radius

re

=

6.378

X

10

6

m

(11)

Also,

J

2

is the "oblateness factor"

J

2

=

1.083

X

10-

3

(12)

and

P

is LOEF's orbital period (in

'

seconds).

Note also, however, that the orbital period itself is slightly time dependent, owing to

orbit contraction due to atmospheric drag. That is,

h

=

h(t)

in (10) anel, moreover,

P(t)

=

21r

[re

+

h(t))3

(13)

where

(14)

is Earth's gravitational constant.

We shall assume

i

=

constant

(15)

i.e., we shall assume that there is no nutation of the LOEF orbit norm al , and take

i

=

24°.

3.5

Time Dependenee of

TJ

The true anomaly of LOEF in its orbit about Earth is the most rapidly changing angle.

Assuming a circular orbit (as before), we have

21r

"I

=

"10

+-t

P

where "10 is a reference value and the orbital period

P

was given, just above, by (13).

(16)

U T I A

S

Time Dependence of Geometrical Factors

3.6

Time Dependenee of

À

Of the seven angles were introduced in §2, naIilely,

we

now come to the last angle, namely

À.

This is the location of a particular experimental

test

patch on the LDEF surface. We shall assume nominally that

(17)

In

particular, ). is constant.

U T I A S

4

Conditions for Shadow

If

the LOEV spacecraft is in Earth's shadow (i.e., is in eclipse), no solar radiation faUs on

any part of LOEF

-

including the test patch. If LOEF is in sunlight, the patch mayor may not

see the sun depending on self-shadowing. In tbis section, we consider the (time-dependent)

geometrical conditions for Earth-shadowing and self-shadowing.

4.1

The Geametrical Canditian far Earth-Shadawing (Eclipse)

Consider Figure 15. The unit vector

03

is shown, as is

oS.

Recall that

~

is along the local

-

-

-vertical ("up") of the spacecraft, and that

oS

is the unit vector from Sun. For a very-Iow-altitude

-satellite like LOEF, it is reasonable to neglect altitude in comparison with Earth's radius in

developing an eclipse criterion. This leads to

as the condition for eclipse.

~.

2

3

>

0

~

LDEF in Eclipse }

oS·

~ ~

0

~

LDEF in Sunlight

-To calculate

oS·~,

we note (2.2) for

oS,

and we have a similar relation for

~:

...

. . .

....

Therefore,

27

(1)

(2)

(3)

Conditions tor Shadow

S

--1

Sunlight

-_.IJ-:

.

--...---I--~---LDEF Orbit

Eclipse

"'--,

---~

-_.

_---Figure 15:

The geometrical condition for shadowing from Earth (eclipse).

U T I A S

Conditions

tor

Shadow

with

Cab, Cbe,

C

ep

and

Cpo

given, respectively, by (2.3), (2.4), (2.11) and (2.12). (Some

transposing is, of course, necessary.)

We could just let the computer calculate

_...

oS·

~

from (3), but an explicit form can be

developed. It can be shown that

~. ~

=

{cos ncos

TJ -

sin 0 cos i sin

TJ} cos(TJe -

ne)

+

{[sin 0 cos

TJ

+

cos 0 cos i sin

TJ]

cos

ie

- sin i sin

TJ

sin ie} sin(TJe -

Oe)

(4)

The time dependences of the angles

have been given in §3.

We complete this discussion of Earth shadowing (eclipse) of the vehicle by defining the

function

(5)

Thus, when

I.Pv

=

1, the vehicle is in sunlight; when

I.Pv

=

0, the vehicle is in shadow.

4.2

The Geometrical Condition for Self-Shadowing

Consider Figure 16. The unit vector 0 is shown, as is

_...

_...

oS.

Recall that 0 is normal to the

_...

test patch on the spacecraft, and that

_...

oS

is the unit vector from Sun. Then the condition that

sunlight fall on the test patch is that LDEF not be in eclipse and that

_...

oS·

_...

0

<

O .

Thus, in combination with the results of §4.1, we have the following criterion for sunlight

falling on the patch:

-+

LDEF in Eclipse

l

-+

LDEF in Sunlight, but Patch in LDEF's Shadow

-+

LDEF in Sunlight, and Patch in Sunlight

(6)

oS •

~ ~

0

and

oS·

0

>

0

... -+-+

~

.

~ ~

0

and

oS·

0

~

0

The inner product

_{... ...}

oS·

~

has been given explicitly above, by (4) .

U

T

I A

S

Conditions

tor

Shadow

30

It

lnside"

"Outside"

"

o

4

Figure 16: The geometrical condition for self-shadowing.

To ca1culate

s·

iJ,

we again note (2.2) for

oS,

and we have a similar relation for

a

from

-+ -+ -+ -+

(2.13):

(7)

Therefore,

(8)

with

_{Cab, Cb}

_e

, Cep

and

Cpo

given, respectively, by (2.3), (2.4), (2.11) and (2.12).

We could just let the computer calculate

....

oS·

0 from (8), but an explicit form cao be

Conditions

tor

Shadow

developed: it can be shown that

~

.

~

= { -

cos 0 sin

"I

sin

À -

sin 0 cos

i

cos

"I

sin

À

+

sin 0 sin

i

cos

À}

cos(

"Ie -

Oe)

{

-cos

ie

sin 0 sin

"I

sin

À

+

+ [cos ie cos 0 cos i -

sin ie sin ij cos

"I

sin

À

- [cos ie cos 0 sin i

+

sin

ie cos ij cos

À

}

sin(~,

-ll,l

(9)

The time dependences of the angles

have been given

in

§3.

We complete this discus sion of self-shadowing by defining the function

, when

s·

iJ

~

0

--

--, otherwise.

(10)

Thus, when

r.p

>

0, the test patch (and therefore also the vehicle) can see Sun; when

r.p

=

I,

the test patch sees Sun flat-on. When

r.p

=

0, the test patch cannot see Sun.

UT' A

S

5

Normalized Solar Heat Flux to Patch

In this section we shall present computer plots of the nonnalized sol ar flux

cp

vs. time,

computed

from the relationships developed in §§2-4. The

detailed time dependenee is quite

complex,

owing to the three simultaneously-aetive time seales identified in § 1.1. We shall in

this seetion

demonstrate that

I.

The "Fast-Time-Seale" component of the flight data (the relatively rapid

oscillations of the order of two or three days) is associated primarily with

the

arbital motion of LDEF around Earth (roughly onee every hour and a

half). Detailed interpretatrion of the eomparison between these computation

and the flight data of

§

1

is made particulary difficult by the sampling rate

(once every 16 hours) used for the latter.

1I.

The

"Slow-Time-Seale"

component of the flight data (the relatively slow

oscillations of the order of 45 days) is associated primarily with the

preces-sianal motion of LDEF's Earth orbit.

lIl.

The "Very-Slow-Time-Seale" component of the flight data (the even slower

oscillations of the order of 300 days) is associated primarily with the

annual

motion of Earth around Sun.

We should also bear in mind, however, that the flight data are

temperatures, not salar fluxes;

the latter can change more rapidly than the fonner. We shall study temperatures more directly

in

§7.

5. 1

Orbital Motion Only

To build our understanding of the

"three-time-scale"

flux history in a logical sequenee of

steps, we begin by eliminating all but the orbital motion of LDEF around Earth. That is, we

temporarily eliminate orbital precession

(n

=

0). We also stop Earth's rotation around Sun

(TI

e

0)

-something

more easily done on a computer than in reality!

The resulting fl ux history is shown in Figure 17. At fust glance, this plot looks both

"black"

and "boring," but closer examination reveals th at it is neither. The only reason the

32

Normalized Solar Heat Flux to Patch

0

.

30

-

+-

- - + - - - t - - - - t - - -

+-

- - - t - - -

- t -

-

--t-

-0.25

--+-

- - - t - - - - + - - - - t - - - -

+-

- - - t - - -

-t-

-

-

-

-t-

-0.20

+ + t t + t t t

-0.15

0.10

0.05

0.00

-0.05

_--1-_ _ _

L -_ _ _ ----L _ _ _ ---1..-_ _ _ L -_ _ ---L _ _ _ --'--_ _ _ --'--_ _

days

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

Figure 17: Normalized solar flux, over ODe year, witb DO LDEF orbital precessioD, aDd

witb DO rotatioD of Eartb arouDd SUD.

plot appears to be "black" is that, while each orbit takes only ...:... 93 minutes, the solar flux

has been plotted over 1 year -

i.e., over 5600 orbits. The variations

within

each orbit are

simply lost when the horizontal scale is compacted to this degree -

by a factor of

103.1.

The details throughout the individual orbits are more like those shown in Figure 18. For

about half the time, LDEF is in eclipse and

<p

=

O. For the other half of the time, the patch

is in self shadow. Thus, reminiscent perhaps of atoms tuming out to be mostly empty space,

<p

=

0 about 75% of the time in Figure 17! For the other 25% of the time,

r.p

is a piece of a

sinusoid whose amplitude can be found from (4.4), (4.9) and (4.10).

Under current assumptions,

~. ~

=

casTf ,

_...

s .

a

= -

sin

Tf

sin

À

Thus, from (4

.

5) and (4.10),

(00 00 23 5° 0° ')4°

100) _ {

sin

Tf

sin

10°

<p

,

,

•

,

,~

,Tf,

-

0

,

~ ~

Tf

~

11";

, otherwise.

(1)

(2)

and the maximum flux is bounded by sin

À

=

sin

10°

=

0.174 ... ,

in agreement with Figure

18.

U T I A S

0.30

0

.

25

0

.

20

0.15

0

.

10

0

.

05

0.00

-0.05

Normalized Solar Heat Flux to Patch

- - -

-I t--I - I - I - t--

-

t-- t-- I -t - i I -

-

t-- ,.-

t--I

I

0.00

0.50

1.00

1.50

2.00

F

i

gure

18:

Normalized solar flux -

over the first two days ooly - with

00

LDEF orbital

precessioo

aod with

00

rotatioo of Earth arouod Suo.

5.2

Orbital

Motion,

Plus

Precession

To continue

building our understanding of the

"three

time scale" flux history in a logical

sequence

of steps, we now

add

the second time scale -

the orbital precession

(0

=1=

0). We

continue,

however,

to

suppress Earth's rotation around Sun

(ije

=

0).

The resulting

flux

history

is shown

in Figure 19. This plot looks "black" (because the

details within

the

5600

orbits have been compressed) but no longer quite so "boring." Orbital

precession causes a slow variation (about every

50

days) in the amplitude of cp, whose details

with

each orbit are again not available from Figure 19.

From

(4.8), under current assumptions,

1"1

=

l(sil1

n

sin i) cos À

±

Jcos

2

n

+

_cos

2

_i

_sin

2

_n

sin

ÀI

(3)

where

the

wide caret indicates

the maximum amplitude achievable in any one orbit.

Taking further the maximum amplitude achievable throughout a precessional cycle, we

have

-1"1

=

Isin

i cos À

+

cos i sin

À

I

=

Isin(

i

+

À)

I

(4)

(The maximum can be shown to occur when

n

=

~.)

Since we are currently assuming

i

=

24°

and

À

=

10°, this implies that

sin(i

+

À)

=

sin 34°

=

0.559 ... , in agreement with

U T I A S

34

---

--

--

---

----

-Normalized Solar Heat Flux to Patch

0.70

+ + + + + _ _ + + 1

-0.65

--+---t---+---t---+---+---+---+----0.60

+ j + + + + + +

-0.55

---+-

___

----~l__---I-_a_---+-_a_----+______o ___ ----+__a,___--_+_____a_----+__-

~-0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

35

-0.05

__ --'--______

L - -_ _ _ _ - - - ' -_ _ _ _ _ _ ---L-_ _ _ _ _ _ - ' - -_ _ _ _ ---L _ _ _ _ _ _ ----L--_ _ _ _ _ _ --L--_ _ _

days

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

Figure 19:

Normalized solar flux, over one year, with LDEF orbital precession, but with

no rotation of Earth around Sun.

Normalized Solar Heat Flux ta Patch

0.30

--~---~--

---r---~---+---0.25

--~---~----

---r---~-'-'1r~-r-r~;r;r~---0.20

--~---~~-H4r4-+-+-~~~~~~~~~~~~~~~~~~---0.15

--~~~~~~~~Hh~~~~~~~~~~~~~~-&-&~~~~~---0.10

---HI~~~i~HII~HIlHa~~~~~~~~~a1~~~~~,*~~~~~~--­

O. 05

---H'fHl.HJHIIH-n-flI--+u--Hl-HHffH'H1It-HIt-HIH-IIHtlt-tfHtlHfl-ttl--tlt-Ht---Ht-~~H_~

...

~~rt_---0

.

00

--~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~---36

-0.05 ---'---'---'---'---...-

days

0.00

0.50

1.00

1.50

2.00

Figurc

20:

Normalized solar flux -

over the first two days only -

with LDEF orbital

precession, but with no rotation of Earth around Sun.

Figure 19.

Just as we expanded Figure 17 for the first two days (as in Figure 18), we can similarly

expand Figure 19 for the first two days (as in Figure 20). Onee again, due to Earth-shadowing

and

self-shadowing, the flux is zero for at least 50% of the time and the remaining half-eycle

sinusoid is also partially truneated.

Narmalized Salar Heat Flux ta Patch

5.3

Earth-Shadowing vs. Self-Shadowing

To further illucidate the relative contributions of Earth-shadowing

vis-à-vis

self-shadowing,

further examination of Figures 19 and 20 will now be made.

5.3.1

Self-Shadowing Only

Figure 21 is for the identical situation as Figure 19 -

with the important exception that the

Earth-shadowing (eclipse) condition has been removed. (The planet Earth has artificially been

made of transparent glass.) Expanded versions of Figure 21 are shown in Figures 22 (Day 0

to Day 2) and 23 (Day 10 to Day 12). Figure 22 should be compared with Figure 20.

Figure 22 is further expanded by a further factor of 10 in Figure 24. The fast-time-scale

(orbital) component, with self-shadowing only, is simply a truncated sinusoid. The amplitude

of this sinusoid, of course slowly changes, as shown in Figure 21.

U T I A S