Earth's Orbit

Let's begin with the most basic fact we
have about Earth's relationship with the sun - Johannes Kepler's first law of
planetary motion. As Earth goes around the sun it traces an ellipse in space,
albeit a slightly imperfect one. In order to describe that ellipse we have a set
of numbers that we plug into a basic description of how to construct the orbit
or, more to the point, how to construct a scale model of the orbit for
astrogational purposes. Because those numbers change over time I must specify
that I have used the numbers valid for the epoch J2000 (that is, the numbers as
they were on 2000 Jan 1.5 (AD2000 Jan 01 AM 12:00 (noon)), Julian Day 2,451,545.0). Those numbers comprise:

1. The Semimajor Axis

a = 149,597,885.651 kilometers

Δa=-74.8
kilometers per century

That number tells us the distance from the geometric center of the ellipse to one of the apsides (either the perihelion or the aphelion). Because astronomers define the astronomical unit by way of a hypothetical circular orbit of period equal to Earth's, that figure is about fifteen kilometers larger than the AU (that is, a=1.00000011 AU).

The change in the semimajor axis comes
from Earth's interactions with the other planets and with the sun's deviations
from perfect sphericity. Thus we get a change of
Δa=-0.00000005.

2. The Eccentricity

e = 0.01671022

Δe=-0.00003804
per century

This number tells us the degree by which
the orbit differs from a circle (e=0), the degree to which it is "stretched
out". Again, as with all the other orbital parameters, the change comes from
gravitational interactions of Earth with the sun's imperfections and with the
other planets.

3. The Inclination

i = 0

Δi=-46.94
arcseconds per century

In addition to its size and shape, we need to know the orbit's orientation in space. The inclination tells us the angle between the plane in which Earth's orbit lies and some standard reference plane; in the case of the solar system the Ecliptic plane provides that reference. We define the Ecliptic plane as comprising that set of points touched by a straight line drawn from Earth's center through the center of the sun to "infinity" in the course of one revolution of Earth on its orbit. That plane coincides with the plane of Earth's orbit, so the inclination of that orbit must equal zero.

But if we define the inclination of the
orbit to equal zero, how can we explain the non-zero change in the inclination?
In this case the change reflects the twisting of Earth's orbit by interactions
with the other planets, whose orbits do not lie in the Ecliptic plane, and the
sun, whose equator does not lie in the Ecliptic plane, but actually represents
the amount by which we must correct the celestial latitudes of the stars in
order to maintain the convenient fiction that Earth's orbit defines zero
inclination for all time.

4. Longitude of the Ascending Node

Ω= -11.26064 degrees (degrees West)

ΔΩ=-18228.25 arcseconds per century

=-5.0634 degrees per
century

If a plane containing an orbit crosses a reference plane, we call the points where the orbit crosses the reference plane the nodes of the orbit and the straight line passing through those points, the line on which the two planes cross each other, we call the line of nodes. Of an orbit's two nodes we take the node where the orbiting body crosses the reference plane from south to north as the ascending node.

We define the directions in the coordinate systems we impose upon our reference plane by analogy with the cardinal directions on Earth. We take north to point in the direction of the orbiting body's angular momentum; that is, from a point north of the orbit we would see the body revolving about the sun in the counterclockwise sense. If we could stand on the sun's north pole, we would see Earth moving in the direction that we define as east.

Because the plane of Earth's orbit coincides with the Ecliptic plane, it has no clearly defined nodes. We define the nodes of the orbit with reference to the nodes of the Celestial Equator: the line of nodes of Earth's orbit consists of that straight line passing through the center of the sun that lies parallel to the straight line passing through the two points where Earth's Equator crosses the Ecliptic plane. Then we define the ascending node of Earth's orbit as the point on the sky where the sun, in its apparent annual motion, crosses the Celestial Equator from south to north, the point that we identify as the First Point in Aries, the Vernal Equinox, the point whose crossing by the sun's center marks the beginning of the first day of Spring.

We take the Ecliptic plane to define Ecliptic latitude, the minimum angle subtended at the sun by the plane and the straight line extending to the object under study. On that basis we can also define Ecliptic longitude as the coordinate marked by the set of planes perpendicular to the Ecliptic plane that all pass through the center of the sun. We have no obvious and natural direction that we can use to define the zero of Ecliptic longitude, so we take that freedom and define the meridional plane extending to the First Point of Aries to mark the prime meridian of the Ecliptic coordinate system, the zero of Ecliptic longitude.

However, that definition comes with a
built-in problem. The slow wobble of Earth's axis, its precession due to Earth's
gravitational interaction with the moon and the sun, makes the First Point of
Aries shift westward at the rate of 18,228.25 arcseconds per century. Since
astronomers first defined it two plus millenia ago, the ascending node of
Earth's orbit has drifted far into Pisces. We must take that drift into any
account we offer of astronomical objects over long elapses of time.

5. Longitude of the Perihelion

ω= 102.94719°

= 1.796767421 radians

Δω= 1198.28 arcseconds per century

=0.00580943 radians per
century

We measure this angle eastward from the
First Point of Aries to the perihelion of Earth's orbit from a vantage point at
the center of the sun (in theory, of course). This angle will increase due to
the eastward shift of the perihelion at the rate of 1198.28 arcseconds per
century (due to Earth's gravitational interactions with the other planets and
with the oblateness of the sun and due to the same General Relativistic effects
that shift the perihelion of Mercury's orbit eastward by 43 arcseconds per
century).

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In addition to size, shape, and
orientation in space, Earth's orbit has a temporal aspect. That aspect gives us
three more numbers:

6. The Orbital Period

P = 365.256898326 days

This number tells us the amount of time that Earth requires to return to any given point on its orbit.

GO TO: Appendix - The Perpetual Calendar

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7. Time of Perihelion Passage

T = AD 2000 Jan 03 PM
10:29

This is the time (Greenwich Mean Time/Universal Time; measured at Longitude Zero on Earth beginning at midnight) when Earth's center crosses the perihelion of Earth's orbit.

GO TO: Appendix - Calculating Perihelion Passage

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8. Orbital Angular Momentum

L
= 4,452,990,073 kilometers squared per second

Actually, this number represents Earth's
orbital angular momentum per unit mass. Subject to a fundamental conservation
law, it does not change measurably unless Earth has a disastrous encounter with
another body of comparable mass. We see that conservation law reflected in the
fact that Earth's mean orbital speed of 29.771 kilometers per second varies from
30.272 kilometers per second at perihelion (r=147,099,302 kilometers or 0.983298
AU) to 29.278 kilometers per second at aphelion (r=152,102,753 kilometers or
1.016744 AU).

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Dynamics of the Orbit

In the conventional deduction of the description of a gravity-shaped orbit we assume that a small body (Earth) of very small mass m suffers acceleration only from the gravitational field of a heavily massive body (the sun) of mass M floating at the origin of our coordinate system. For this derivation we use polar coordinates in a frame so oriented that we can describe the small body's motion with only two spatial variables - radial distance r from the origin and longitude θ measured from an arbitrarily chosen radial line. In that system we have from Newtonian dynamics two equations that describe the forces that act upon the small body.

For the radial component of the force we
have

(Eq'n 1)

The second term in that equation represents the centrifugal force, the force with which a body resists moving in a circle: in this case it diminishes the rate at which the small body accelerates toward the larger body. For convenience I have divided the mass of the small body out of the equation, so strictly speaking the equation represents accelerations and not forces.

The longitudinal component of the force
gives us

(Eq'n 2)

The second term in that equation
represents the Coriolis force, another obligatory force (like the centrifugal
force) that appears automatically in descriptions of curved motion. That
equation encodes the law of conservation of angular momentum, from which we
obtain

(Eq'n 3)

In order to obtain an equation that
describes the shape of the orbit, we want to eliminate the time variable from
those equations. We start by reformulating Equation 1 to see whether we can get
the time variable away from the radial distance. We know that

(Eq'n 4)

for a body traveling both radially and
longitudinally, so we have

(Eq'n 5)

If we solve Equation 2 for d^{2}θ/dt^{2}
and substitute the solution into that equation, we get

(Eq'n 6)

That substitution tacitly incorporates
conservation of angular momentum into that equation. I also applied Equation 4
again in the first term on the right side of the equality sign. Working through
the differentiation in the first term on the right and substituting yet again
from Equation 4 in the second term gives us

(Eq'n 7)

At this point we have moved the time variable away from the radial distance. Now I want to eliminate it altogether.

Equation 3 tells us that dθ/dt=L/r^{2},
so Equation 7 becomes

(Eq'n 8)

Now it looks like we have the
differentials, not of the radial distance, but of its inverse, the curvature of
the trajectory. To explore that implication let's carry out the following
differentiations:

(Eq'n 9)

and

(Eq'n 10)

Solving Equation 10 for **
M ^{2}r/M**2

(Eq'n 11)

We can use that result and the expression
for dθ/dt
from Equation 3 to rewrite Equation 1. The radial acceleration equation thus
becomes

(Eq'n 12)

Multiplying that equation by -r^{2}/L^{2}
gives us then

(Eq'n 13)

As usual, we approach that equation with
a trial solution based on a function whose form remains unchanged by a double
differentiation,

(Eq'n 14)

That trial solution transforms Equation
13 into

(Eq'n 15)

We see immediately that we must have

(Eq'n 16)

because both terms are constants, and

(Eq'n 17)

We can eliminate
θ_{0}
from our solution through an appropriate orientation of our coordinate grid and
we know that we want only the real-number part of the solution, so we now
rewrite Equation 14 as

(Eq'n 18)

We now want to determine the mathematical form of K.

Consider the specific energy of the small
body; that is, the small body's total energy (vis-a-vis the large body) divided
by the small body's mass. When the small lies a distance r from the large body
we have

(Eq'n 19)

We can represent the specific kinetic
energy as a function of the small body's angular velocity and write

(Eq'n 20)

Using Equation 3 to represent the angular
velocity as a function of the specific angular momentum, we rewrite that as

(Eq'n 21)

in which I used Equation 9 to convert the
**Mr/M**2.
Replacing 1/r from Equation 18 gives us then

(Eq'n 22)

That result and Equation 18 allow us to
rewrite Equation 19 as

(Eq'n 23)

And we can solve that equation readily
for K;

(Eq'n 24)

Finally we invert Equation 18 and write
it as

(Eq'n 25)

in which

(Eq'n 26)

and

(Eq'n 27)

Equation 25, describing the path that the small body traces around the large one, also describes a conic section of eccentricity e. We use the eccentricity as if it were a variable parameter in order to identify the four basic types of conic section:

1. if e=0, then the small body traces a circle about the large body,

2. if 0<e<1, then the small body traces an ellipse,

3. if e=1, then the small body traces a parabola, and

4. if e>1, then the small body traces an hyperbola.

As I noted before, Earth traces an
ellipse as it goes around the sun.

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The Temporal Aspect

More often than not, astronomers express the law of conservation of angular momentum in the form of Johannes Kepler's second law of planetary motion, the statement that the radius vector extending from the sun to a given planet sweeps out equal areas in equal times (see Item #8 above). We can exploit that law to devise an equation that will relate the longitude of Earth on its orbit to the time that has elapsed since the planet's last perihelion passage.

We must use integration to obtain a
suitable description of an elliptic segment partly bounded by an arc of the
planet's orbit, so we take the minuscule element of area to consist of an
isoceles triangle whose sides have length r and whose base has length rdθ,
so we have

(Eq'n 28)

For the length of the radius vector I
have used Equation 25 in which I made the substitution r_{0}=(1+e)r_{p},
in which r_{p} represents the radius of the orbit at perihelion.
Kepler's law gives us dA/dt=L.
If we multiply that equation by the minuscule element of time, integrate both
sides of the resulting equation, and divide the result by
L,
we get

(Eq'n 29)

Thus we have the numbers and equations
with which we describe Earth's orbit.

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