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ORBITAL MECHANICS FOR THERMAL ENGINEERS

Bob Poley

20 November 2002

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WHY ARE WE DOING THIS?

• Thermal engineers need to know enough about orbits to understand where the sun and earth are during the orbit, and how that might change over time.

• We also need to know how the spacecraft is oriented in the orbit.

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WHY ARE WE DOING THIS?

• Then we can begin to answer questions like– Is there a shady side?– If I have a radiator, does it need to be

shielded from the sun? From the earth?– Where should I locate a battery?

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CLOSED ORBITS ARE ELLIPTICAL

• A circular orbit is just a special case.

• Open orbits can be parabolic or hyperbolic, but you could only fly around on it once.

• The body you are orbiting is located at one focus of the ellipse.

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ELEMENTS OF AN ELLIPSE

B

B

AF FO

A

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EQUATIONS FOR THE ELLIPSE

• Distance OA is the semi-major axis, a

• Distance OB is the semi-minor axis, b

• With the origin at O, and the x-axis along the semi-major axis, then

1b

y

a

x2

2

2

2

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EARTH ORBITS

• Let the earth be at the focus, F, with radius of earth =

• Let the altitude of apogee =

• Let the altitude of perigee = ahbh

2/hhRa baE

a2RhRh EbEa

ER

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ORBITAL PERIOD

• The period is the time it takes to go around the orbit once. By integrating equations resulting from Kepler’s second law, it can be shown that the period, P is

• where = 5.166 x /a2P 3

2312 hrkm 10 /

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ECCENTRICITY

• Eccentricity, , is defined such that distance OF = a. Eccentricity = 0 for a circular orbit.

• It can also be shown that

21ab

aba 22

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TIME OF FLIGHT

• A circular orbit is easy. The time it takes to get from one point on the orbital path to another point is proportional to the angle between the two points.

• Time = angle x period / 360• Any point can be used as a baseline.

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TIME OF FLIGHT

• An elliptical orbit is trickier, and we have to define true and eccentric anomaly.

• Perigee is usually used as the baseline.

• Let the satellite be at point P.

• The true anomaly is the angle , measured from perigee.

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TIME OF FLIGHT

• If we draw the reference circle shown in the next slide, true anomaly, , is related to the eccentric anomaly, E, which is measured from the center of the reference circle to point P'.

cos1

cosEcos

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ECCENTRIC AND TRUE ANOMALY

FOA

E

P

P'

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TIME OF FLIGHT

• Then the time t, from perigee, is given by (using radians)

EsinEa

t3

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TIME• One day = 24 hours, measured from when to

when?• Suppose we measure between successive

highest elevations of the sun. This is an apparent solar day.

• But with this definition, no two days would be identical because the earth moves faster at periapsis (December) than it does at apoapsis (June).

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TIME• Define a mean sun, in which the earth is in a

circular orbit. Then the mean sun moves along the celestial equator at a constant rate of 360 degrees in 365.25 mean solar days.

• A mean solar day has 24 mean solar hours.• Suppose we measure relative to the fixed stars;

say the vernal equinox. This is a sidereal day.• A mean solar day is slightly longer than a

sidereal day (24h 3m 56.5+ s).

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TIME

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COORDINATE SYSTEM• For earth-orbiting spacecraft, the most

common coordinate system is the geocentric-equatorial system. Its origin is the center of the earth. The fundamental plane is the equatorial plane. It is inertial, and the earth rotates with respect to it.

• +X is in the direction of the vernal equinox (first point of Aries), and +Z is in the direction of the North Pole.

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CLASSICAL ORBITAL ELEMENTS

• Five independent quantities are sufficient to describe the size, shape and orientation of an orbit. These are– a, the semi-major axis , the eccentricity– i, the inclination , the right ascension of the ascending node , the argument of perigee

• These are illustrated in the next slide.

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CLASSICAL ORBITAL ELEMENTS

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INCLINATION• In the geocentric-equatorial system mentioned

above, inclination is simply the angle between the orbit plane and the equatorial plane.

• More precisely, it is the angle between the K vector (North Pole normal to the equatorial plane) and the positive orbit normal (right hand rule).

• This avoids ambiguity when the inclination is near 90 degrees.

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RIGHT ASCENSION• The ascending node of an orbit is that point in

the orbit where the spacecraft passes through the equatorial plane headed north.

• The right ascension is the angle, measured clockwise in the equatorial plane, as viewed from the North Pole, from the first point of Aries to the ascending node.

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ARGUMENT OF PERIGEE• The argument of perigee is the angle,

measured in the orbital plane, from the ascending node to perigee.

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BETA ANGLE, • The beta angle, , is the angle between the

solar vector and the orbit plane. If the solar vector is in the orbit plane, = 0. Beta can go to 90. The general convention is that is positive when the sun is on the same side of the orbit plane as the positive orbit normal (right hand rule).

• TRASYS does not follow this convention; TSS does.

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VARIATION OF BETA will vary over time, as the sun moves

relative to the right ascension of the ascending node of the orbit.

• The sun moves in the ecliptic plane, which is at 23.5 with respect to the equatorial plane.

• The sun makes a full circuit in 365.25 days, although the rate of motion is not constant.

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VARIATION OF BETA• The orbit precesses; that is, the right

ascension () varies as time passes.• If the orbit is not sun-synchronous, will

move through the range (23.5 + i).• Gilmore gives the equation for :

)icos(sin)sin()isin(cossin sss

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VARIATION OF

icosnp

RJ

2

32

22

22

2

22

31isin

2

31

p

RJ

2

31

an

21ap

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WHAT’S ALL THIS, THEN?

• J2 is a dimensionless constant with a value of 0.001082626. It is related to the Earth equatorial oblateness. This is the difference between the equatorial and polar radii, about 22 km.

• More about this later, when we get into perturbations.

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CAN WE SIMPLIFY THIS?

• (1 - 1.5 sin2 i) goes from 1 (at i = 0) to 0 (at i = 54.74) to -0.5 (at i = 90), so the maximum value for this term is 1.

• For a circular orbit, = 0, so p = a, and a = R + h, so the max value of R/a = 1.

• The term in brackets is now very close to 1:

• Call it 1.

001623910010826051111J511 2 .)].)(.([))((.

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MORE SIMPLIFICATIONS

• So for a circular orbit,

icosaa

RJ5.1

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2

2

icosRa

RJ5.1

3

5.3

2

icosa

R

Ra

RJ5.1

3

3

32

2

2

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SUN-SYNCHRONOUS 1

• In a sun-synch orbit, we want the orbit to precess 360 in a year, or 365.25 days.

• This sets the value for at 360/365.25 deg/day.

• We can now solve for the inclination, i, which is a function of a.

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SUN-SYNCHRONOUS 2

)icos(hR

R)B)(A(

5.3

E

E

3

23

)km16.6378(

sec/km0.398603)0010826.0)(5.1(A

rad

deg180

day

sec86400B

daydeg/96399.9AB

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SUN-SYNCHRONOUS 3

• Plug h and RE into this equation and solve for i:

• This will get you pretty close, but if you needed extreme accuracy, you can use this value of i to re-evaluate the term in brackets

• and recompute i as necessary

)icos(00.1hR

R098919.0

5.3

E

E

isin5.11hR

RJ5.11 2

2

E

E2

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COMPUTING

• So finally, having computed the inclination, you can use FINDB6 or Satellite Tool Kit to determine as a function of time and to determine the extreme values of .

• It is often true that the coldest orbit will be when = 0 (or its smallest absolute value) and that the hottest orbit will be when is such that the orbit is just fully sunlit.

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AND SHADOW TIME

• The orbit will be fully sunlit when

hR

Rsin

E

E1

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AND SHADOW TIME

• For a circular orbit, the shadow points can easily be calculated by hand. If is the orbital angle, measured from the dawn terminator, then

180in 360out

cos

cossin sunfull1

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AND SHADOW TIME

into shadow

sun

= 0Dawn terminator

out of shadow

= 90Sub-solar point

= 270Middle of shadow

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FINDB6

• To run this program, you need to know starting and stopping dates, universal time, altitudes of apogee and perigee, inclination, right ascension, and argument of perigee.

• For sun synch orbits, ALWAYS start at the vernal equinox (so you know where the sun is). The date and universal time for this is best found in the Astronomical Almanac.

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FINDB6

• A sun synch orbit will usually be described by its equatorial crossing time. Hence, a 10 AM ascending orbit, etc.

• If you’re given a descending orbit time, add 12 hours to get ascending time.

• For FINDB6, the right ascension is then given by RA = 180 + (crossing time, hr) x (15 / hr)

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GEOSYNCHRONOUS 1

• Geosynch orbits are easy from an orbital mechanics point of view, but hard from a thermal management point of view.

• They are circular with a 24-hour period, so the altitude must be 35863 km (5.623 earth radii).

• Their inclination is 0, so they have a range of 23.5

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GEOSYNCHRONOUS 2

• If the satellite is earth-oriented, then the only plausible surfaces for dumping heat are the north- and south-looking faces.

• If you do a 180 roll twice a year (at the equinoxes), you can always keep the sun off one of these faces.

• The solar arrays will undoubtedly be panels which extend from these faces, so keep that in mind.

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PERTURBATIONS

• Most of the preceding equations are based on two-body theory, where the mass of each body is assumed to be located at a point, or at least symmetrically spread in spherical shells.

• We don’t generally need to worry about the effects of the sun, moon, or other bodies on a earth-orbiting satellite, but the mass distribution is another story.

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PERTURBATIONS

• Typically for an earth satellite, the effect of the sun is about 0.6 times the effect of J2, the effect of the moon is three orders of magnitude down from the effect of J2, and so forth. Bate, Mueller, and White, page 11, has a table of these effects.

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PERTURBATIONS

• J2, as mentioned earlier, is a result of the earth’s oblateness. There is also J3, caused by the polar bulge. The height of the bulge at the North Pole is 16.5 meters, so the value of J3 is about 3 orders of magnitude smaller than J2, and can thus be ignored for our calculations.

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PERTURBATIONS

• There is J22, which is a second tesseral harmonic related to the ellipticity of the Earth equatorial plane and is responsible for the long-term (860-day) resonance effects on geo satellites.

• Other perturbations are caused by solar pressure, atmospheric drag, tidal effects, outgassing (or cryo expulsion), etc.

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SPACECRAFT ORIENTATION

• A spacecraft most often maintains a constant orientation as it goes around the orbit.

• Earth oriented means that the spacecraft always has the same face pointed to the earth. Earthshine is thus constant throughout the orbit.

• Space oriented, or solar inertial, means that the spacecraft always has the same face pointed toward the sun.

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SPACECRAFT ORIENTATION

• Earth oriented is typical for earth resources missions, or communication satellites.

• Space oriented is typical for a mission like Kepler, where you want to view a particular star for a long period of time.

• These orientations can be combined, as in MTI, where we flew solar inertial until we wanted to take data or downlink. Then we shifted to earth oriented.

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QUICK CHECKS ON FLUXES

• We often want to do hand calcs, or checks of TRASYS or TSS outputs.

• For earthshine, this is pretty easy, especially if you’re earth oriented. Use QUICKVIEW to get the geometric view factor, F, from the surface to the earth. Then

)ttanconsearthshine)()(F)(A(QES

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QUICK CHECK ON ES

• In this, A is area, is emissivity, and I get the earthshine constant from page 2-6 of Gilmore’s book. For earth oriented, this will be constant throughout the orbit. Otherwise, you should pick known positions where it’s easy to figure out the orientation between the surface of interest and the earth.

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QUICK CHECK ON AL

• For albedo, it’s only a little more complicated. At the subsolar point, albedo is a maximum, and it goes to more or less zero over the terminator, and of course when looking at the backside of the earth. At the subsolar point,

))(costtanconsDS)(ttanconsAL)()(F)(A(QAL

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QUICK CHECK ON AL

• If we wanted an orbital average value for an earth oriented orbit, we’d have to integrate with respect to the orbital angle, (sorry about the nomenclature). Let Q be the value of the absorbed albedo at the subsolar point. Then

Q2dsinQ2

d

dsinQQ

2/

0

0

0

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QUICK CHECK ON DS

• For an earth-oriented spacecraft, the position of the sun relative to any one of the sides varies over the orbit, so we have to find out how the angle between the normal to the surface of interest and the solar vector changes. Fortunately, this is not too hard.

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SUN ANGLE VARIATION

• Define a coordinate system moving with the spacecraft where +z is away from the earth, +y is in the direction of travel, and +x completes the right-hand system. Then with the direction cosines of the surface normal as l, m, and n, and as the angle between the sun and the surface normal

sinlsinncosmcoscos

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SUN ANGLE VARIATION

• If cos is negative, then the sun is on the back side of the surface. Remember that this equation doesn’t include shadowing by other surfaces or by the earth.

• Then at any given orbital position,

• To get an orbital average, this would have to be integrated with respect to the orbital position angle, , as we did for albedo.

))(costtanconsDS)()(A(Q surfDS

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ANGLE TO THE LIMB

• If we’re trying to size earth shields, as for a cryo radiator, we sometimes need to determine the angle from nadir to the limb of the earth.

• You can do the geometry, but it turns out to be exactly the same angle as the angle for a fully sunlit orbit.

hR

Rsinangle

E

E1

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LAGRANGE POINT ORBITS

• The Lagrange points are those points where the gravitational pulls of two bodies (like sun-earth or earth-moon) are balanced.

• There are 5 such points, shown in the next chart. Three are unstable (L1 through L3) and the other two are stable.

• JWST will be located at L2, so that a single shield blocks both earth and sun.

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LAGRANGE POINT ORBITS

Sun

Earth

L1

L2

L5

L3

L4

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REFERENCES

• FUNDAMENTALS OF ASTRODYNAMICS, Bate, Mueller, and White, 1971, Dover

• MISSION GEOMETRY; ORBIT AND CONSTELLATION DESIGN AND MANAGEMENT, Wertz, et al, 2001, Microcosm Press & Kluwer Academic Publishers

• ORBITAL MECHANICS, Chobotov, ed., 1991, AIAA Educational Series

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REFERENCES

• ASTRONOMICAL ALMANAC, annual, US Govt Printing Office

• EXPLANATORY SUPPLEMENT TO THE ASTRONOMICAL ALMANAC, Seidelman, ed., 1992, University Science Books

• SATELLITE THERMAL CONTROL HANDBOOK, Gilmore, ed., 1994, Aerospace Corp Press