{ Practical astronomy | Astronomy | The Sun and the Earth's orbit }

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The Sun and the Earth's orbit

The physical Sun

The Sun is a star. It is the star that Earth and the other planets orbit, therefore it is about 300000 times nearer than the nearest other stars.

In ancient times astronomers distinguished (fixed) stars and (moving) planets. The Earth was neither, because it was down here, while stars and planets were "up there". Sun and Moon were planets, as were Mercury, Venus, Mars, Jupiter and Saturn.

Today we prefer physical definitions over phenomenological ones. A planet now is something that orbits a star. That makes Earth a planet, but Sun and Moon are no longer planets. A star is an object that generates energy from nuclear fusion, which it radiates into space as mostly visible light. That is exactly what the Sun does.

The Sun is basically a lump of interstellar matter that came together under its own gravity to form a much denser ball of gas. Such gas is then at a high temperature, cannot maintain neutral molecules, and is a fully ionised plasma. Initially, almost 75% are hydrogen and almost 24% are helium, with 2% heavier elements thrown in. As we have seen with the Earth's atmosphere, the upper layers, under gravity, press down on the lower layers. Further inside, the pressure has to be higher, but density and temperature will also be higher. This counteracts gravity and prevents the whole star from collapsing into a black hole.

If the total mass of the gas ball is sufficient, then the temperature and pressure at the centre are high enough to start nuclear fusion of hydrogen into helium.

An interstellar cloud would have very low density, and the density would fizzle out gradually with distance from the cloud centre. In a star like the Sun there is a sharp boundary between very low density in the surrounding volume and high density at and just below what we are then justified to call the surface. For the Sun this is 696000 km from the centre.

Core, radiative zone, convective zone

At the centre of the Sun the temperature is 15.7 million K and the pressure of 25.5·10¹⁵ Pa is 250000 million times higher than at Earth's sea level. Even at the bottom of our ocean, 10 km down, the pressure is only about 1000 times higher than at the surface of the ocean.

In these extreme conditions, the nuclei of the hydrogen atoms cannot stay apart and a fraction of them combine into the heavier helium nuclei. The mass balance of this nuclear reaction is such that the helium nucleus is a tiny bit lighter than the sum of its parts. This small mass has been converted into light (plus the odd speeding neutrino) according to Einstein's mass-energy equivalence E = m·c².

Hydrogen fusion in the Sun is a gradual process. But over a few thousand million years, the effect on the composition of the gas is quite dramatic. At the centre of the Sun, the fraction of helium is now enhanced from its original 24% to about 60%; almost half of the original hydrogen has already gone.

Initially, the energy is released as gamma rays. But these do not make it far. They encounter gas particles and mostly act to maintain the heat at the centre. As much heat as is generated, will also seep out through the layers of gas toward the surface. The first 500000 km of the way, this happens through radiation: hot gas emits black-body radiation, this is absorbed again by the gas, the gas remains hot and re-emits, and so on. Overall, slowly, this allows heat to transfer outward. At about 70% of the solar radius, the gas is sufficiently cool that it becomes too opaque for the radiation. The bottom of this layer has to take all the heat, which then swells up to the surface in a convective motion: Hot bubbles move up and cooler surface material moves down to fill the gap.

Photosphere, chromosphere, corona

The convection has to end at the surface, since there is virtually no gas above to carry the heat. Instead, the surface emits black-body radiation into space, in accordance with its temperature of three times that of a hot electric plate. This makes the surface visible, and we call it the photosphere as the location from where the sunlight, as we receive it, is emitted.

At the surface, the temperature is still 5800 K. However, the gas is almost neutral, with only 3% of atoms ionised. The photosphere is actually tens to hundreds of kilometres thick with reduced temperature higher up. This leads to the limb appearing darker than the central part of the solar disc on our sky. At the centre we look straight to the hotter bottom of the photosphere, but near the limb we look both along a longer light path but to a lesser depth: we see a less hot surface, which therefore appears less bright.

The Sun does not end at the photosphere. Above lies a layer of 500 km thickness called the chromosphere, where the temperature reaches a minimum of 4100 K. This layer is quite prominent in the hydrogen Hα emission line and therefore has a deep red colour. Above the chromosphere is the corona. In the transition the temperature rises again, to a million K. This is possible, because the density is very low, which makes the gas easy to heat and difficult to cool.

Nonetheless the supply of such energy into the corona is not trivial. It requires strong magnetic fields and the fast flow of plasma along these. This leads to dynamic arcs of higher density in the corona, which are seen in Hα either as bright prominences above the solar limb or as dark filaments in front of the solar disc (the chromosphere and photosphere).

Magnetic field, rotation and sunspots

The convective motion below the photosphere, combined with the rotation of the Sun, leads to that rotation being faster at the equator and slower toward the poles. (This is the same Coriolis force that on Earth makes high- and low-pressure areas spin one way or another.) On the other hand, the rotating Sun is a dense ionised plasma, where magnetic field lines have to follow the motion of the gas. Thus the solar magnetic field will change from a global bipolar field of 0.0002 T into one that is wound up and magnified at mid latitudes. There it can twist and form loops of 0.3 T field strength that extend out from the convective zone into the corona above. Although the field, broadly speaking, has to follow the motion of the dense plasma, the strong local field does somewhat suppress the convective motion. This leads to spots of somewhat cooler surface temperature (3000 to 4500 K), which then appear darker than the surrounding photosphere. These are the sunspots. They appear in groups or active regions that have a bipolar east-west structure, corresponding to the two places where the loop of magnetic field lines intersects the photosphere. Small spots come and go in a matter of hours, but groups and their principal spots survive for days or weeks.

There is an approximately 11-year cycle of change in the number and complexity of sunspots. At the cycle minimum there are often no spots at all, and the magnetic field is quite ordered and pointing north-south. Over a few years, the field lines get wound up, first at higher latitudes, in later years of the cycle more towards the solar equator. These are the latitudes where local magnetic loops and sunspots form. The east-west direction of the magnetic field is opposite between the northern and southern hemisphere. Toward the end of the activity cycle the latitudes of activity merge at the equator and the magnetic field gradually returns to a weak polar field, now of opposite polarity than 11 years earlier.

Solar ephemeris

Ephemerides are ephemeral data describing the motion of a celestial body. Simply put, they are a table of times, and the celestial position and distance at that time. Ephemeris and ephemerides are the singular and plural of the word, but ephemeris are also not really countable, so that both terms are used interchangeably and essentially have the same meaning.

As Earthlings we need solar ephemeris to tell us where in our sky the Sun will be at a given time. But the movement in question is actually that of the Earth around the Sun. Even more correct would be to say they orbit around each other or around their common centre of mass. That is a helpful view for binary stars of vaguely similar mass, but for a relatively tiny planet orbiting an overwhelmingly massive star the image in our head can justifiably be that of the planet orbiting the star. The orbit integration gives us the position of one object w.r.t. the other and makes no judgement whether one of the two objects is itself moving or not. The calculations are correct in spite of our slightly simplified image of what is going around what else.

Gravitation

Newton realised in 1687 the single reason for both the proverbial apple falling to the ground and the Moon moving around the Earth. Every mass (the apple, the Earth, the Moon) attracts every other mass with a certain force, the gravitational force

F = G · M · m / r²

G = 6.674 · 10⁻¹¹ m³ / (kg s²)

The force is a vector, and it being attractive, the vector points from the mass "feeling" the force to the other mass. Thus it points "inwards" along the radius vector r that connects the two masses. This is a central force.

Newton had also established that a force F acting on a mass m will accelerate it, i.e. change its speed v. This has nothing to do with gravity, but essentially states that a massive body will move in a straight line with constant speed, unless forced to do otherwise (pun intended).

F = m · dv/dt

Force and velocity are both vectors. If they are parallel, the force changes the value of the velocity (notionally the length of the vector). If they are orthogonal (perpendicular to each other), then the value of the velocity remains unchanged, but the force changes the direction of the velocity vector.

When the apple is released from the tree it enters free fall, initially with zero velocity. Earth's gravity will accelerate it at 9.81 m/s². It will therefore start to move towards the ground and will move faster and faster the longer the fall takes. When it hits the ground one side of the apple will experience a severe compressive shock from the collision with the ground.

There is energy involved here. Hanging from the tree there is a potential for the apple to fall and speed up. Movement is kinetic energy to the tune of E = m·v²/2. Before the fall begins, this energy is a potential energy.

Newton had eventually realised that the Moon is not so different from the apple. Gravity by the Earth applies in the same way, though the distance r is much larger. The crucial difference is that the Moon is not hanging around waiting to fall. It evades the fall my moving sideways. From the start, the Moon has a significant velocity v that is more or less perpendicular to the radius vector r from Earth to the Moon. The gravitational force will then not so much change the value of velocity, but its direction. Depending on the balance between force toward the Earth and velocity sideways, the Moon may end up making a circle around the Earth without ever colliding with it.

Why do we talk about the Moon and not the Sun? Because the mental step from apple to Moon is simpler than that from the apple-Earth system to the Earth-Sun system. But the force ruling the movement of the Earth around the Sun is exactly the same.

Circular motion in gravity

Consider the graphic, and the Sun (yellow) and the Earth (blue) with masses M and m. Let the radius vector r point from the Sun to the Earth, and let the Earth move with velocity v perpendicular to the radius vector r. Consider a small time interval dt passing. The Earth will have moved a small amount dx = v·dt sideways (upward in the graph), but the distance r will not have changed measurably. We can say that the Earth has moved a small angle dx/r w.r.t. the Sun. For the Earth to make a circular orbit around the Sun, the condition is that in the time interval dt the force has also changed the direction of the velocity by that same angle. That is to say, there should now be a small component dv to the velocity vector that is perpendicular to the original movement. But this is just so much to make the vectors v and r orthogonal again. Starting with the equality of the two angles we can substitute variables, until we have the expression for the circular velocity at a given distance from the Sun:

dv/v = dx/r
(dv/dt) / v = v / r
(dv/dt) = v² / r
(dv/dt) m = m v² / r
F = m v² / r
G M m / r² = m v² / r

v² = G M / r

This is the condition for circular motion. But also, the vectors have to be orthogonal. In the graphic this is the black orbit.

Faster than circular motion

While it is mathematically possible to have circular motion, the idea that the actual velocity exactly matches the requirement, both in how fast and in being directed perpendicular to the force, is unphysical; it would be a hugely improbable coincidence.

Suppose the body under solar gravity moves perpendicular to the force, but faster than circular. This is the situation of the planet being in perihelion. It will then move to a larger distance from the Sun, and the velocity vector will also point slightly outward. The latter leads to a slowing down of the planet, which leads to a loss of distance from the Sun. That leads to a speeding up and increase of distance. And so on. The orbit is elliptical and larger than the circular orbit. In the graphic this is the red ellipse.

I is remarkable that the orbit is a closed curve. This happens only with a central force that also diminishes as r². There is then one point in space where perihelion occurs and aphelion occurs in the exactly opposite direction.

In energy terms, at perihelion the planet has excess kinetic energy. It then gains distance from the Sun, which converts kinetic into potential energy. At aphelion the process reverses: There is a lack of kinetic energy, but the loss of distance accelerates the planet and increases its kinetic energy again.

This picture of a start at perihelion and of elliptic motion breaks down if the perihelion velocity is more than √2 times the circular velocity. In that case solar gravity fails to gain the upper hand over the initial kinetic energy and the planet will just fly away into infinity (though not beyond and only after an infinite amount of time, and other stars not intervening). The "orbit" is then not elliptic, but hyperbolic. A parabolic orbit occurs in the case that the velocity is exactly √2 times the circular velocity.

We have heard about these things relating to the Earth and spacecraft. Earth's mass is such that a few hundred km above the surface, the circular velocity is 7.9 km/s. This is the speed a rocket must reach to deliver a satellite into orbit. 11.2 km/s is the speed required to send a space probe beyond Earth's gravity into the solar system.

So much for faster than circular motion. What if the planet is slower when on the right in the graphic? Then this location is not its perihelion, but its aphelion. It is too slow to maintain its current distance from the Sun and moves inwards along a smaller ellipse, which is blue in the graphic.

Deviations from the ellipse

Although gravity is exactly (as far as we can measure) an r² force, the elliptic orbit is not exactly true. There are three reasons for this:

1. Sun and Earth are not alone. Other bodies in the solar system tug a tiny bit on both these bodies to spoil the ideal image of two bodies alone in the universe.
2. Although gravity acts over very large distances, it does not do so instantaneously. The force on the Earth points not to the "current" position of the Sun, but to where the Sun was a few minutes earlier, in line with the distance and the speed of light. Because we use light to see the Sun, naturally we see the Sun where it was 8 min ago. But gravitational force, or any physical effect, is also limited to the speed of light.
3. Special and general relativity speak of strange effects on space-time when things move fast and when large masses are involved. These effects also slightly invalidate simple Newtonian gravity.

The ellipse is still a very good approximation of the orbit, but there will be slow changes to the orientation, shape and size of the ellipse. For reasonably good ephemeris, such change of the parameters of the ellipse have to be taken into account. This is still far simpler than to calculate in detail and in tiny time steps the gravitational effects that all the planets have on each other, in a relativistic space-time.

Calculating ephemeris is also called integrating the orbit. To calculate where a planet is at a given time, we need to know its track in space, meaning its orbit around the Sun. We also need to know the time dependence of its location within the orbit. The orbit can be thought of as a geometrical object, while the timely movement along the orbit is up to the physical process of gravitational attraction. Although the geometry of the orbit is also determined by the physics, in practice the maths becomes more manageable by splitting the problem.

A circular orbit

The simplest and most symmetric shape of an orbit compatible with Newtonian gravity is a circle with the Sun at its centre. In the case of the Earth the plane of the orbit is the ecliptic, which we have already defined as one of several coordinate systems. The additional orbital elements we need are

a

Radius of the orbit. We express this in Gm, but might also note that for the Earth this is the astronomical unit and a distance unit in its own right.

n

Mean daily motion. If we imagine the observer at the centre of the orbit, the orbiting object will move by 360° during the completion of one orbit. n simply tells us how many degrees per day the object moves on average. This number is equivalent to the period of the orbit. In the case of a circular orbit, the actual motion at all times is equal to this mean motion. The Earth completes one orbit in one tropical year.

These are the values for the Earth's orbit around the Sun, the astronomical unit and the equivalent of the tropical year:

a = 149.597870 Gm

n = 360° / 365.242190 d = 0.985647359°/d

The time unit we use is the day TT. UT is unsuitable, because it is adjusted to follow the rotation of the Earth and does not run evenly.

What is still missing is a zero point or integration constant. This is the knowledge of the position at one point in time. For a circular orbit of the Earth around the Sun it is convenient to use the time of a vernal equinox. The vernal equinox is not only a place in the sky (where ecliptic and equator cross each other), but also an annual event (when the Sun moves through that place on the sky).

We can look up a time (TT, not UT) of equinox in an astronomical yearbook and we know at that time that the geocentric ecliptic longitude of the Sun is zero (USNO et al. 1998a, p.A1 and p.K9)

T₀ = 2000-03-20T07:35 UT
ΔT(T₀) = 66 s
T₀ = 2000-03-20T07:36 TT
T₀ = JD 2451623.816667

λ_hel(T₀) = 180°

Note the twist here: At the vernal equinox, by definition, the geocentric longitude of the Sun is zero. But we integrate the orbit of the Earth around the Sun, and the heliocentric longitude of the Earth at the vernal equinox is 180°.

Integration of the orbit is quite simple for the circle. The ecliptic spherical coordinates at an arbitrary time t are simply

λ_hel(t) = 180° + n · (t − T₀)
β_hel(t) = 0
r(t) = a

To convert from heliocentric to geocentric, we change the longitude by 180° and negate the latitude

λ(t) = n · (t − T₀)
β(t) = 0
r(t) = a

How wrong is this orbit? Longitude and distance will have an annual variation of the error. Due to our integration constant, the longitude is correct at each vernal equinox, but at each autumn equinox the circular orbit will be 3.75° ahead of the truth. This is enough to make a sundial about 15 min fast, and about as much as the field of view of a pair of binoculars. The distance error might be considered negligible; its relative error is 0.017%, which is the actual eccentricity of the orbit.

The Kepler orbit

Although possible in principle an exactly circular orbit will not exist in reality. Kepler described the actual situation in three statements, the Kepler laws:

1. The planets move in ellipses with the Sun in one of their focal points.
2. The radius vector of a planet encompasses equal areas in equal time intervals.
3. The squares of the revolution periods of two planets relate to each other like the cubes of their orbits' semi-major axes.

The language is very technical and terse, but most will become clear below. The third law is in essence our criterion for circular motion. If we convert v²∝r⁻¹ via the size of the circular orbit into a condition on the orbital period, we get P²∝r³. The third law also implies that the orbital period depends only on the semi-major axis and not on the eccentricity of the ellipse.

The second law is the most complex. Physically, it expresses the conservation of angular momentum. For ephemeris calculation it describes the progress of the planet along the ellipse as time passes. It says, for example, that the planet moves faster near perihelion and slower near aphelion, and it says by how much. So it is quite a powerful statement, but converting it into an orbit integration algorithm is not trivial. The Kepler equation is a mathematical translation of the second law; it states

E(t) − e · sin(E(t)) = n · (t − T_P)

Here T_P is the time of the previous perihelion. We know n already, and t is simply the time. e is the numeric eccentricity and describes how much the ellipse differs from a circle. E is the eccentric anomaly and will help us calculate the Cartesian coordinates of the planet in the orbital plane.

Note that the Kepler equation mixes angles and trigonometric functions. There is then potential for confusion over the units. e has no dimension and the result of the sin() function has no dimension. Therefore the angles E and n must also be without dimension, i.e. expressed in radian and not in degrees.

Kepler's first law appears to say very little. A subtlety often missed is that an ellipse is a planar geometrical object. In a complex force field the planet might have to follow a path that is not restricted to one plane. Kepler in effect says that gravity is a central force. Another thing is that the ellipse is a closed curve: The planet will return to the same place after a time interval of one revolution has passed. Kepler in effect says that gravity is a 1/r² force. The first law has also one other vital piece of information: The Sun is not at the centre of the orbit, it is offset and sits in one of the two focal points.

What then of these focal points, where are they? This is where the eccentricity e comes into play. We convert a circle into an ellipse as follows:

• We choose a radius a for the circle.
• We choose an eccentricity e between 0 and 1.
• We choose a direction for the x and y axes in the plane of the circle.
• We contract the circle along the y axis by a factor b/a = √(1 − e²).
• Along the x axis the ellipse has radius a, just like the circle. We call a the semi-major axis. Along the y axis the radius b is shorter, more so, the larger e.
• In addition to the centre C there are now two distinct focal points F and S that are a distance e·a either side of the centre.
• The Sun occupies one of the focal points S. We call the points where the ellipse intersects the x axis the perihelion P and aphelion, for these are the points of the ellipse nearest to and farthest from the Sun.
• We now place the planet at an arbitrary point T of the ellipse. We note the corresponding point X on the circle. It has the same x coordinate value and a y value larger by a factor a/b.
• We note the angle PCX, which is the eccentric anomaly E.

The Kepler orbit (of the Earth in the plane of the ecliptic) has these elements:

a

Semi-major axis (in Gm or au).

e

Numeric eccentricity (dimensionless). This describes how much the ellipse deviates from the encompassing circle. e=0 is the circle, e=1 would be the extreme case where the ellipse opens up into a parabola.

n

Mean daily motion (in °/d).

ω

Argument of perihelion (in °). This is the angle between the vernal equinox and the perihelion and describes the orientation of the ellipse in the ecliptic.

The integration constant (the position at a particular time) can be found in the literature in one of three forms:

• The time T_P of perihelion, when X and T coincide and E = 0.
• The mean anomaly M at a standard epoch T₀.
• The mean longitude L at a standard epoch T₀.

Do not be confused by the words or the fact that the mean anomaly and longitude appear to be angles. They are just parametrisations of time. Depending on which of the integration constants you find, the Kepler equation takes a different form:

E(t) − e · sin(E(t)) = n · (t − T_P)

E(t) − e · sin(E(t)) = n · (t − T₀) + L(T₀) − ω

E(t) − e · sin(E(t)) = n · (t − T₀) + M(T₀)

The first form is the most logical, while the third form is the most practical. In practice we proceed like this. If L(T₀) is given, calculate

M(T₀) = L(T₀) − ω

Else, if T_P is given, use

T₀ = T_P
M(T₀) = 0

In any case, then calculate the mean anomaly

M(t) = M(T₀) + n · (t − T₀)

Now the Kepler equation is definitely

E − e · sin(E) = M

Algebra cannot solve this for E. But with programmable electronic computers, we can solve this brute-force by iteration.

E₀ = M
E_i+1 = M + e · sin(E_i)

We can abort the iteration when successive values become virtually identical, e.g. closer than 0.02":

|E_i+1 − E_i| < 0.0000001 rad

The iteration does not require many steps, so this excessive precision will not cost us much time. Thus we have calculated the angle E from the time t. We can now calculate quite simply the Cartesian coordinates

x_C = a · cos(E)
y_C = a · sin(E) · (1 − e²)^0.5
z_C = 0

These, however, refer to the centre of the ellipse as origin. The heliocentric coordinates are

x_S = a · cos(E) − a · e
y_S = a · sin(E) · (1 − e²)^0.5
z_S = z_C

Even then the axes are oriented to the ellipse rather than the vernal equinox. To obtain ecliptic coordinates, we have to rotate the xy-plane by the angle −ω.

x_hel = x_S cos ω − y_S sin ω
y_hel = y_S cos ω + x_S sin ω
z_hel = z_S

To convert from heliocentric to geocentric, we negate each Cartesian coordinate.

x_geo = −x_hel
y_geo = −y_hel
z_geo = −z_hel

The orbit can be described for different coordinate systems. Natural would be that the algorithm delivers heliocentric coordinates for the mean ecliptic and equinox of date (EOD). However, sometimes the elements in the literature instead yield heliocentric coordinates for the mean ecliptic and equinox of J2000. We may have to add one or two further coordinate transforms to get the coordinates we need.

Another issue is that in reality the orbit slowly changes. One set of elements then describes the orbit of the planet at a particular time, and this time is usually stated alongside the elements.

We can use the following Kepler elements to describe the orbit as at J2000 and to obtain ecliptic coordinates for the mean equinox and ecliptic of J2000 (Standish and Williams, 2018a):

a = 1.00000216 au

e = 0.01671123

n = 35999.37244981° / 36525 d

ω = 102.93768193°

L(T₀) = 100.46457166°
T₀ = JD 2451545.0

A Kepler orbit would be correct only if there were only the Sun and the one planet involved, and if general relativity had only negligible effects. In reality, the mutual gravitational attraction of all the planets affects the motion of each, so that none of them runs in a precise, everlasting ellipse. The planets from Jupiter outwards are particularly sensitive to these effects.

So how wrong is the J2000 ellipse as orbit? Over a 10-year period, the longitude error is very small, 0.005° or 20"; this is not a clear annual cycle. The distance error is also small at about 0.005 Gm or 0.003%. Over 100 years the errors double or treble to 0.01° and 0.015 Gm. In 1000 years they would grow another tenfold.

We have to bear in mind, though, that the inner planets up to Mars suffer less perturbation than the outer planets. Looking for good calculations for all planets, the Kepler ellipse would show much larger deficiencies for Jupiter, Saturn and Uranus.

The Simon et al. algorithm

The algorithm published by Simon et al. (1994a) was adopted by the International Astronomical Union into its SOFA software library (SOFA 2016a). This algorithm takes the corrections of the Kepler orbit rather further. The Kepler elements are given a modification that is quadratic in time, but then also quite complex periodic terms are applied to the mean longitude and semi-major axis for all planets. The accuracy is at worst less than 2'.

We now have to introduce two further elements of the Kepler orbit, which we always need for non-Earth planets. The Simon et al. algorithm allows the orbital plane to change over time, which introduces an inclination of the Earth's orbit against the J2000 ecliptic. To describe that situation, two new angles are required: the inclination i of the orbit against the ecliptic and the longitude Ω of the ascending node (where the orbital plane intersects the ecliptic). The argument of the perihelion ω then takes on a slightly different meaning, in that it measures the angle from the ascending node to the perihelion, not from the vernal equinox. The full list of Kepler elements then is:

a

Semi-major axis (in Gm or au).

e

Numeric eccentricity (dimensionless).

n

Mean daily motion (in °/d).

ω

Argument of perihelion (in °). This is the angle between the ascending node and the perihelion and describes the orientation of the ellipse in the orbital plane.

i

Inclination (in °). This is the inclination angle between the ecliptic and the orbital plane.

Ω

Longitude of ascending node (in °). This is the angle between the vernal equinox and the ascending node (where the orbital plane intersects the ecliptic).

In the literature, more often than not, the perihelion is located by a slightly different parameter than ω. This is denoted by the symbol π or ϖ and called the longitude of perihelion. That parameter is simply

π = ϖ = ω + Ω

Adding those two angles makes no sense, and so this quantity has no precise geometric meaning. You best always think and calculate in ω, but be aware that the numbers others give you will be ϖ.

Remember that there is also the integration constant, i.e. the specification of one point in time and the position at that time. We listed above the three possibilities, time of perihelion, mean anomaly M at a standard epoch, and mean longitude L at a standard epoch. With the introduction of an orbital plane that does not coincide with the ecliptic, the meaning of L changes in line with the change of meaning for ω. Before, with the orbit in the ecliptic and Ω = 0, we had M = L − ω, but now we need to use

M = L − ω − Ω = L − ϖ

The two new elements i and Ω mean that two further coordinate rotations have to be applied after the one by −ω:

x₁ = x_S cos ω − y_S sin ω
y₁ = y_S cos ω + x_S sin ω
z₁ = z_S

x₂ = x₁
y₂ = y₁ cos i − z₁ sin i
z₂ = z₁ cos i + y₁ sin i

x_hel = x₂ cos Ω − y₂ sin Ω
y_hel = y₂ cos Ω + x₂ sin Ω
z_hel = z₂

x_geo = −x_hel
y_geo = −y_hel
z_geo = −z_hel

Physical ephemeris

By ephemeris is primarily meant the position of a celestial body at a given time. Physical ephemeris would describe not where but how the body appears to us. For the planets and Moon these parameters are

• the apparent magnitude,
• the apparent radius,
• the elongation from the Sun, zero for the Sun itself,
• the phase angle, zero for the Sun,
• the illuminated fraction of the apparent disc, one for the self-luminous Sun,
• the orientation of the body as seen from Earth, i.e.
  • the position angle of the rotation axis,
  • the inclination of the rotation axis toward the observer,
  • the central meridian.

The inclination and central meridian are also the heliographic latitude and longitude, resp., of the centre of the solar disc.

At unit distance of 1 au the Sun has an apparent V magnitude of −26.70 (Unsöld and Baschek 1999a, p.181) and universally apparent magnitude drops with the square of distance. Since magnitude is 2.5 times the decadic logarithm of the brightness, apparent magnitude then drops as 5 times the logarithm of the distance.

V = −26.70 + 5 lg(r / au)

The apparent radius is similarly simple to calculate, given the radius of the Sun of 696,000 km (USNO et al. 1998a, p.K7):

ρ = asin(0.696 Gm / r)

To calculate the orientation of the Sun and its state of rotation we use the expressions of Davies et al. (1996a) for the celestial coordinates of the solar pole of rotation and the argument of the prime meridian W:

T₀ = JD 2451545.0 d

α₁ = 286.13°
δ₁ = +63.87°
W = 81.1° + 14.1844°/d · (t − T₀ − r/c)

The inclination i of the rotation axis towards the observer, the position angle PA of the rotation axis on the observer's sky, and the central meridian CM can then be calculated (USNO et al. 1998a, p.E87):

sin(i) = −sin(δ₁) sin(δ) − cos(δ₁) cos(δ) cos(α₁−α)

sin(PA) = sin(α₁−α) cos(δ₁) / cos(i)
cos(PA) = [sin(δ₁) cos(δ) − cos(δ₁) sin(δ) cos(α₁−α)] / cos(i)

sin(K) = [−cos(δ₁) sin(δ) + sin(δ₁) cos(δ) cos(α₁−α)] / cos(i)
cos(K) = sin(α₁−α) cos(δ) / cos(i)
CM = K − W

Solar eclipses

About twice a year the Moon passes between the Sun and the Earth. Since the apparent sizes of Moon and Sun are almost identical, an observer on Earth may find part or all of the Sun temporarily invisible, this is called a solar eclipse. In such an eclipse, the Moon casts its shadow toward the Earth. This shadow comes in two parts:

1. The central umbra is a conical volume behind the Moon where no direct sunlight enters. It shrinks with distance from the Moon and the vertex of this cone is approximately where the Earth is. If Moon and Earth are close enough, there is a small area on the Earth's surface (at most about 150 km in diameter) where the Sun is completely covered by the Moon. This is a total solar eclipse, which lasts at most about 4 min.
2. The penumbra is a conical volume that widens behind the Moon. Here some, but not all sunlight enters. To the observer on Earth, part of the Sun is hidden behind the Moon, so this is called a partial solar eclipse.

By these definitions it would be a partial eclipse, when the Moon lines up exactly with the Sun, like in a total eclipse, but the Moon is too far from Earth to cover the Sun. The observer would see a ring of sunlight around the Moon, so this is called an annular solar eclipse.

If we consider the eclipse as a global event, then we can look up maps that show where on Earth we can see what type of eclipse. In this frame of mind there are the following types of eclipse:

• In a total solar eclipse there is a region on Earth, about 100 km wide and thousands of km long, where the umbra passes and causes darkness for a few minutes. Observers within this zone of totality will see a total eclipse for those few minutes and a partial eclipse for about an hour before and after. Observers outside the zone of totality may see a partial eclipse, unless they are too far away from the zone and see no eclipse at all.
• In an annular solar eclipse the zone of totality is replaced by the zone of annularity. Observers in this zone see a ring of sunlight around the Moon for a few minutes. They see a conventional partial eclipse before and after. Outside the zone things are just like in a total eclipse.
• The distinction between total and annular eclipse is whether the Moon is close enough to the observer to cover the Sun. In a hybrid solar eclipse the answer is "no" for observers where the eclipse occurs near sunrise or sunset, but the answer is "yes" for observers where the eclipse occurs near midday.
• In a (globally) partial solar eclipse the umbra misses the Earth and passes above the northern or southern polar region. There is still a considerable chunk of the Earth where the Moon covers part of the Sun for an hour or two.

From this remains unclear how different a total eclipse is from the other types. The Sun is completely obscured by the Moon, but by a very small margin. This allows us to see the faint chromosphere, prominences and corona of the Sun. This is a spectacle that many amateur astronomers travel around the world to see. Even if 99% or 99.9% of sunlight is blocked in a partial eclipse, the result is not the same. By comparison, the partial (or annular) eclipse is entirely underwhelming.

Predictions for solar and lunar eclipses, as well as transits of Mercury or Venus across the Sun, can be obtained from Espenak (2018a).