Sunday, November 02, 2008

Compute Orbits with Radius Angles

PURPOSE: Determine equation for ellipse given a series of angles but not ranges from the focus.
REASON: Earth bound instruments can readily determine angles but not so readily distances. Many instruments (i.e. sextants) can point to distant celestial objects, and telescopic instruments can point to them more accurately to produce more precise angular measurements. However, range determining instruments (such as radar) are much more limited; thus, distances often have to be deduced from a series of angular measurements.



Initial Observation
ObservationObserved
DateAngle
(same time)(degs)

07/07/07

14.03

(Assume same time of day for subsequent observations.)



For convenience, we'll use pointer from focus (Sol) to perihelion as reference ray to determine angular displacement. To do this, we must introduce two artificialities:

  1. This assumes that we know where the perihelion is. Of course, if we knew position of perihelion, then we'd essentially know the orbit. Forgive us this artificiality, we'll explain later.
  2. This further assumes that we're able to translate observed geocentric angles to heliocentric angles. This will also be further explained later.






Delta Angle
ObservationObservedDelta
TotalDateAngle (ν)Angle (Δν)
Days (t)(same time)(degs)(degs)

0

07/07/07

14.03

Start

10

07/17/07

36.00

21.97

JPL Horizons Web SiteΔν= νi - νi-1

Average angular velocity (ω) between any two observations is readily determined by dividing angular displacement by duration between observation times. For example, above table would give us an initial ω of 2.2° per day.







Angular Velocity

is computed in both degrees per day as well as radians per second.
ObservationObservedDeltaAngular
TotalDateAngle (ν)Angle (Δν)Velocity (ω)
Days (t) (same time)(degs)(degs)(°/day)(Rad/Sec)
007/07/0714.03StartStart

Start

1007/17/0736.0021.972.20

4.44 x 10-7

2007/27/0754.7718.771.88

3.79 x 10-7

JPL Horizons Web Siteνi - νi-1Δν/Δt
ω * π
180*86,400 sec/day
tΔνωω'

Three observations enables computation of two successive angular velocities as shown above. We note a significant difference between the two ω's; thus, an elliptical orbit is indicated.

That's too bad, because if the ω's were the same, we could determine the period by dividing them into total degrees per orbit (360°). In next table, we'll determine a column of "false periods" by wrongfully assuming circular orbits.









Point Estimates of Period

ObservationObservedDeltaAngularFalse
TotalDateAngle (ν)Angle (Δ ν)Velocity (ω)Period
Days(same time)(degs)(degs)(°/day)(Rad/Sec)(Days)
007/07/0714.03StartStart

Start

Start
1007/17/0736.0021.972.20

4.44 x 10-7

163.9
2007/27/0754.7718.771.88

3.79 x 10-7

191.8
3008/06/0770.1115.341.53

3.10 x 10-7

234.7
JPL Horizons Web Siteνi - νi-1Δν/Δt
ω * π
180*86,400 s/d
T ' = 360°/ω
tΔνωω'

Period can be readily determined from a constant angular velocity as shown above. However, above periods are "false" because the computed angular velocity is not constant; they constantly vary, and any one value applies only for one of an infinity of radius vectors within a yet to be determined range.

  • If the computed ω was constant, then the orbit would be circular and the period would be "true".
  • If the computed ω happened to be the orbit's average angular velocity, then the period would be fortuitously true. Of course, this could only happen to at most, one particular value. It certainly couldn't happen to the multiple values shown in above table.

For the few rows in above table, the ω is ever decreasing which indicates corresponding radii are ever increasing (recall Kepler's Laws). Thus, corresponding periods can lead to guesses for corresponding radii. (See next table.)



….







Initial Radius Estimates

We know the Initial Radius Estimates will need future adjustment.
Because we don't yet know "a", semimajor axis; our first guess must come from the formula for
circular orbit periods.
Thus, our initial guess is meant to be a ballpark estimates probably not so accurate.

ObservationObservedDeltaAngularRadius (from Sol)
TotalDateAngle (ν)Angle (Δ ν)Velocity (ω) Initial Estimate
(Days)(same time)(degs)(degs)(°/day)(rad/sec)(Meters)(AUs)
007/07/0714.03StartStart

Start

StartStart
1007/17/0736.0021.972.20

4.44 x 10-7

0.877 x 10110.58
2007/27/0754.7718.771.88

3.79 x 10-7

0.974 x 10110.65
3008/06/0770.1115.341.53

3.10 x 10-7

1.11 x 10110.74
4008/16/0782.5412.421.24

2.51 x 10-7

1.28 x 10110.85
JPL Horizons Web Siteνi - νi-1Δν/Δt
ω' * π
180*86,400
(μ/ω2)1/3
R
1.5 x 1011
tΔνω'ωRR'

Radius estimates come from temporarily assuming a circular orbit for each position for which we've computed a point estimate of angular velocity.

V = ω R =
(
μ
R
)1/2

Rearranging above to solve for initial estimate, R:

R =
(
μ
ω2
)1/3

Substituting following values:

  • μ = 13.27 x 1019 m3/sec2
  • 1 AU = 1.50 x 1011 m

Example (40 day row) follows:

R'=
(
13.27 x 1019
(2.51 x 10-7)2
)1/3
R'=
 
1.24 x 1011 meters
 
R =
 
0.85 AU
 



….




Series of Radius Vector Estimates

ObservationObservedDeltaAngularInitial Est.Average
TotalDateAngle Angle VelocityRadiusAngle
Days(same time)(degs)(degs)(rad/sec)(AUs)(degs)
007/07/0714.03StartStartStartStart
1007/17/0736.0021.974.44 x 10-70.5925.0
2007/27/0754.7718.773.79 x 10-70.6545.4
3008/06/0770.1115.343.10 x 10-70.7462.4
4008/16/0782.5412.422.51 x 10-70.8676.3
5008/26/0792.6910.152.05 x 10-70.9887.6
JPL Horizons Web Siteνi - νi-1Δν/Δt(μ/ω2)1/3i - νi-1)/2
tνΔνωRνAve
….



X Component of Vector

Initial Est.AverageX
RadiusTAComponent
Seq(AUs)(degs)(AUs)
10.5925.00.53
20.6545.40.46
30.7462.40.34
40.8676.30.20
50.9887.60.04
61.1196.9-0.13
71.24104.7-0.31
81.37111.3-0.50
91.49117.1-0.68
101.62122.2-0.86
111.74126.7-1.04
121.85130.8-1.21
(T'2μ/4π2)1/3 i - νi-1)/2Ricos(νAve)
tRiνAveXi
….

http://en.wikipedia.org/wiki/Root_mean_square
http://www.easycalculation.com/statistics/root-mean-square.php

http://en.wikipedia.org/wiki/Cubic_equation
http://www.akiti.ca/Quad3Deg.html
Reference Ray: From Sun to point of Aries.
Vernal Equinox
The point on the celestial sphere where the Sun crosses the Earth's equatorial plane from south to north. Also called the first point of Aries.
Because the First Point of Aries is the zero-point for calculating coordinates on the Celestial Sphere, its own coordinates are always fixed, regardless of its motion. They are, of course, zero hours Right Ascension and zero degrees Declination.
The First Point of Aries is the point in the sky where the Celestial Meridian, the Celestial Equator and the Ecliptic all meet. It is presently in the southwest of Pisces, moving slowly towards its neighbouring constellation, Aquarius.
Astronomers use the astrological symbol for 'Aries', depicting the head and horns of a Ram, to indicate the point where the Ecliptic crosses the Celestial Meridian, known as the First Point of Aries.
Derivation
Phrixus and Helle were the children of Athamas the King of Thebes. When their wicked step-mother threatened to have them killed, they escaped on a magical flying ram with a golden fleece. Helle was understandably alarmed by this experience, and fell from the ram into the sea, but her brother Phrixus survived and landed safely in Colchis. There, he sacrificed the ram to Zeus, who promptly placed it among the stars, while the King of Colchis kept its golden fleece.
Stars
Aries is one of the least conspicuous of the zodiacal constellations, and has only two stars above third magnitude. These are Hamal and Sheratan, the Alpha and Beta stars of the constellation. Unusually, these two stars not only appear to be close together in the sky, but actually are: they lie just six light years from one another.
The First Point of Aries
The First Point of Aries is a place of vital importance in the sky - here, the Ecliptic and the Celestial Equator cross, and when the Sun reaches this point, as it does once a year, the Vernal Equinox occurs. The First Point of Aries also marks the Celestial Meridian, which is the zero-point for calculations of Right Ascension.
Because of the effects of precession on the Earth, though, the First Point of Aries moves through the sky, and in fact it left the constellation from which it takes its name in about the year 70 BC, when it entered the neighbouring constellation of Pisces. Nonetheless, it retains the name 'First Point of Aries'. Roughly 23,000 years from now, the Sun will have completed its circuit of the zodiac, and the First Point will once again lie among the stars from which it takes its name.
The Ram
Derivation Phrixus and Helle were the children of Athamas the King of Thebes. When their wicked step-mother threatened to have them killed, they escaped on a magical flying ram with a golden fleece. Helle was understandably alarmed by this experience, and fell from the ram into the sea, but her brother Phrixus survived and landed safely in Colchis. There, he sacrificed the ram to Zeus, who promptly placed it among the stars, while the King of Colchis kept its golden fleece.
Stars Aries is one of the least conspicuous of the zodiacal constellations, and has only two stars above third magnitude. These are Hamal and Sheratan, the Alpha and Beta stars of the constellation. Unusually, these two stars not only appear to be close together in the sky, but actually are: they lie just six light years from one another.
The First Point of Aries The First Point of Aries is a place of vital importance in the sky - here, the Ecliptic and the Celestial Equator cross, and when the Sun reaches this point, as it does once a year, the Vernal Equinox occurs. The First Point of Aries also marks the Celestial Meridian, which is the zero-point for calculations of Right Ascension.
Because of the effects of precession on the Earth, though, the First Point of Aries moves through the sky, and in fact it left the constellation from which it takes its name in about the year 70 BC, when it entered the neighbouring constellation of Pisces. Nonetheless, it retains the name 'First Point of Aries'. Roughly 23,000 years from now, the Sun will have completed its circuit of the zodiac, and the First Point will once again lie among the stars from which it takes its name.
Scorpius includes a broad region of the Galaxy, in the general direction of the Core. This explains the density of the Milky Way in the constellation, and the large number of star clusters
Constellation of Aries is a member of zodiac just west of the Pleiades (M45) in Taurus, an asterism of three stars with second or third magnitude lined up a bit strained marks the head of the Ram. Though Aries is tiny and very faint constellation, that had included the vernal equinox about two thousand years ago, and Aries was an important zodiac as the first member of that.
equinox
Time when the Sun is directly overhead at the Earth's Equator and consequently day and night are of equal length at all latitudes. This happens twice a year: 20 or 21 March is the spring, or vernal, equinox and 22 or 23 September is the autumn equinox.
The variation in day lengths occurs because the Earth is tilted on its axis with respect to the Sun. However, because the Earth not only rotates on its own axis but also orbits the Sun, at the equinoxes the two bodies are positioned so that the line on the Earth separating day from night passes through both of the Earth's poles.
hut(2)
First Point of Aries
The First Point of Aries, also called the vernal equinox point, is one of the two points on the celestial sphere where the celestial equator intersects the ecliptic. It is defined as the position of the Sun on the celestial sphere at the time of the vernal equinox. It was named after the constellation in which it occurred in ancient times - Aries. However due to precession, the point gradually moves around the ecliptic. It entered the constellation Pisces in about 500 A.D. (or around 100 A.D. if the modern constellation boundaries are used) and will enter Aquarius in about 2600 A.D. The other such point, at the autumnal equinox, is the First Point of Libra.
In the equatorial coordinate system the First Point of Aries is defined to have a right ascension of zero. The declination is also zero due to the position on the celestial equator.

It is important to keep in mind the geocentric nature of this definition, and how things can seem backwards when thinking solar-centrically, as when trying to use orbital elements. For instance, Earth reaches perihelion shortly after the hibernal solstice, three-fourths of a year after vernal equinox. With all the vernal-equinox talk, it's easy to think Earth's longitude of perihelion should be about 270�. But longitude 0 is the far side of the Sun at vernal equinox -- i.e. Earth won't reach longitude 0 until the autumnal equinox, and so the longitude of perihelion is more like 90� (about 103� in fact.) Sidereal clocks agree with conventional clocks at the autumnal equinox instead of the vernal for similar reasons.

The time of a complete rotation of Earth relative to a point on the sky is called a sidereal day; this is measured relative to the first point of Aries (the vernal equinox),[ 2] which is a point on the sky that slowly moves as the Earth's rotation axis precesses. The mean value of the sidereal day is shorter than the Julian day by 3 minutes 55.9 seconds. The tropical year, which is the year that the Gregorian calendar is based on, is measured relative to the first point of Aries. The sidereal year is measured relative to a fixed point in space; quasars are used as the reference points.
Positions on the sky are measured in terms of right ascension (RA or α) and declination (dec. or δ). These correspond to longitude and latitude on Earth. The declination is measured in degrees relative to the celestial equator, which is the projection of Earth's equator onto the sky. The declination of the equator is 0°, the declination of the north pole is 90°, and the declination of the south pole is −90°. The right ascension is defined in units of time, with 0 hour at the first point of Aries, and the value of the right ascension at the zenith increasing as time passes. A full circle of the equator corresponds to 24 hours. Right ascension was defined to make finding objects with a telescope easier: the right ascension at the zenith changes by one hour in one hour of sidereal time.
Because the precession of Earth's rotation axis causes the first point of Aries to moves along the equator over time, the right ascension and declination of the stars change with time. To counteract this, astronomers traditionally express the positions of stars for the coordinate system of a particular date. Currently current standard is Epoch 2000, or the coordinate system for January 1, 2000. Epoch 1950 was used previously.

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