Analytical Solutions and a Clock for Orbital Progress Based on Symmetry of the Ellipse

Abstract

Kepler’s discoveries were permitted by his remarkable insight to place the Sun at the focus of an elliptical planetary orbit. This coordinate system reduces a 2-dimensional orbit to a single spatial dimension. We consider an alternative coordinate system centered on the “image focus,” which is the symmetrical (mirror) counterpart of the “real focus” occupied by the Sun. Our analytical approach provides new purely geometric formulae and an exact relationship for the dynamic property of orbital time. In addition, considering the mirror symmetry of the ellipse leads to a simple approximation: the radial hand of an orbital clock rotates counterclockwise at a nearly steady angular velocity 2π/T about the “image focus,” where T is the orbital period. This approximation is a useful pedagogic tool and has good accuracy for orbits with low to moderate eccentricities, since the deviation from the exact result goes as eccentricity squared. Planetary comparisons are made. In particular, the angular speeds of Mercury and Jupiter are highly variable in the geocentric and heliocentric reference frames, but are nearly constant in the image focus reference frame. Our findings resolve whether the image focus is the location for observing uniform motion of an elliptical orbit, and pertain to their stability.

1. Introduction

Kepler’s discoveries played a seminal role in the scientific revolution, and his prescient concepts about orbits and gravity were prerequisite to Newton’s discovery of the law of gravitation [1]. Key to Kepler’s discoveries is the geometry of the ellipse. Although just a flattened circle, the ellipse has such confounding mathematical properties that its perimeter can only be expressed in terms of infinite series or elliptic integrals [2]. Thus, the path followed by an orbiting object has a troublesome length that greatly complicates relating orbital position to time. Yet, spatial position and time are the essential variables of dynamics, and quantifying their mutual relationship is the central problem. Note that, in contrast to ballistic trajectories, or harmonic oscillators, or waves, the variable of time never appears in the familiar equations for orbital position, velocity, kinetic energy, and so on. Even though Kepler’s second law—“equal areas in equal times”—is a remarkable dynamic constraint, it provides a convoluted way for a person to visualize or calculate orbital progress. Therefore, numerical methods are widely used to deduce orbital progress in ideal elliptical and actual solar system orbits [3]. The present study explores analytical representations of the ellipse and perfect elliptical orbits.

Solar system dynamics and evolution are a major research area in the greater field of celestial mechanics [3]. Keplerian elliptical orbits remain important to modern science because these are used in teaching classical mechanics, and are the basis for exploring perturbations of planetary orbits from this ideal [4,5,6]. The largest perturbations arise in interplanetary interactions, but even the force of massive Jupiter on its neighbor Mars at the closest approach is only 0.00017 times the solar force. Hence, analytical expressions for the time-dependence of stable elliptical orbits will further the efforts in planetary science and astronomy, whereas quantifying the static properties of the ellipse is of great interest in mathematics.

Regarding the broad field of celestial mechanics, historic and modern efforts have sought the location where an observer would perceive planetary motions as uniform [7]. The motivation is the ancient philosophical and theological postulate that heavenly bodies move in perfect circles [8] (p. 153). Maeyama [7] states that many astronomy reference books, including Littrow’s ~1837 book as edited by Stumpff [9], consider the “empty focus” to be this location, as it lies on the line between the apsides, thereby meeting symmetry requirements. Since terminology varies, precedence is unclear, and much work has been translated, this paper uses “image focus,” which connotes the symmetry of the ellipse. However, neither the exact location nor the existence of a perfect point for observation have been proven, in part because few have explored elliptical geometry or orbits from the perspective of the “image” focus, see Section 3.3.

To address the above issues quantitatively and exactly, we follow Kepler’s geometrical approach. Our paper is organized as follows. Section 2 and Section 3 explain why Kepler’s choice of the polar coordinate system centered on the Sun provides many key mathematical simplifications. Section 3.2 shows how the mirror image of the Keplerian coordinate system yields several new and surprising results, summarized below. We use only trigonometry, symmetry, and basic calculus so that our improved and new descriptions of ellipses are transparent, and can be used in teaching, as these require little or no familiarity with either vector methods or elliptical functions. Because the topic is solar system dynamics, our formulation assumes that the mass of the controlling, central body is much larger than the masses of each and all orbiting bodies.

In this report, only Kepler’s first law is presumed. Section 4 shows that additional orbital properties and a new geometric constant are revealed by considering the symmetrical coordinate system centered on the “image focus,” which was previously used for series or numerical calculations [7,10,11,12]. Results for velocity and conserved dynamic and geometric quantities are tabulated. Section 5 computes the exact orbital time by using relationships revealed by our image coordinate system. No formula is currently available that directly describes progress along the entire ellipse (see e.g., [13,14]). Section 6 devises an “orbital clock” visualization and analytical approximation that has very good accuracy for all planets and most moons in the Solar System: the radial hand of this clock rotates counterclockwise at a nearly steady angular velocity 2π/T about the “image focus,” where T is the orbital period. Section 7 verifies our equations by through comparison to the planets. Section 8 discusses our results, implications, and pedagogical value. Section 9 concludes.

2. Background

2.1. Kepler’s Laws

Kepler’s Laws of planetary motion are typically paraphrased in the following manner:

• Planets move in elliptical orbits with the Sun at one focus.
• A line between the Sun and a planet sweeps out equal areas in equal times.
• The square of the orbital period of a planet is proportional to the cube of its average distance from the Sun.

As previously noted [15], these statements are difficult to find among the thousands of pages of Kepler’s many books, written in Latin between 1596 and 1621. Their closest representation may appear in the translation of his Harmonice Mundi [16], where Kepler states:

• “the orbit of a planet is elliptical, and the Sun, the fount of motion, is at one of the focuses of that ellipse”
• “the true daily arcs of a single eccentric orbit …. have a proportion to each other which is the inverse of the proportion of the two distances from the Sun”
• “the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances”

These seminal statements are diluted by much other material, including Kepler’s many attempts to relate planetary orbits to the five regular solids, likely motivated because their shapes are “perfect,” and because only five planets were known during his time. Kepler’s efforts to quantify orbital phenomena using mathematics and geometry, and to prove or disprove available hypotheses by precise mathematical analysis of available data, were pioneering steps in the development of the scientific method.

Only Kepler’s first law, that orbital paths are elliptical, will be presumed in our continuation of his geometrical approach. The next sections demonstrate that his other two laws are consequences of his observations and his first law, and that Kepler chose a particularly convenient coordinate system to analyze orbital motions.

2.2. Geometry of the Ellipse

Figure 1 shows the well-known, essential characteristics of this “flattened circle”, whose radial height is less than its radial width. Specifically, the semi-minor axis c, which is half the height of the ellipse, is smaller than the semi-major axis a, which is half of its width. These axial lengths define the eccentricity ε:

$$\epsilon =\sqrt{1-c^2/a^2 }.$$

(1)

Equivalently, c can be expressed as $a\sqrt{1-{\epsilon }^2}$.

The ellipse has two foci, referred to below as the real (occupied) focus and the image focus, that are spaced at distances of ±aε from the geometric center. An essential relationship, evident from the familiar “pins and string” method to construct an ellipse, applies to any arbitrary point m on the orbital path:

2a = r + r_i

(2)

where r and r_i are the radial distances of the orbiter from the real focus and the image focus, respectively (Figure 1).

This study is partially motivated by pedagogy. Hence, we use α, θ and ε, instead of F, f, and e as in celestial mechanics, to describe the “antifocal anomaly,” the “true anomaly,” and the eccentricity, respectively, because F, f, and e are more commonly used to represent forces, functions, and Euler’s number in mathematics and physics. Moreover, α and θ are commonly used to denote angles. We use “image focus” rather than Maeyama’s [7] “empty focus” or Fukushima’s [11] “antifocus.” Our term properly indicates both position and inaction, whereas “antifocus” would be appropriate if this point were an impossible source of gravitational repulsion.

3. Coordinate Systems and Their Interrelationships

Many important problems in physics are most easily visualized and solved in a coordinate system that is congruent with the physical boundaries, symmetry, or some dynamic condition of the system of interest. The following equations all provide exact geometric representations of the ellipse, but use different spatial coordinates and origins. We focus on the essential formulae, their uses, and interrelationships.

3.1. Conventionally Used Coordinate Systems for the Ellipse

The most familiar algebraic representation of the ellipse is expressed in Cartesian coordinates (x,y) measured from its center:

x²/a²+ y²/c² = 1

(3)

where a and c are defined above. Recasting these Cartesian variables in terms of polar coordinates (R, ϕ) about the center of the ellipse provides the coordinate system used in elliptic integrals:

x = RCosϕ,  y = RSinϕ;  R² = x² + y²

(4)

Substituting the Equation (4) relationships into Equation (3), and using the identity Cos²ϕ + Sin²ϕ = 1 yields a direct relationship between R and ϕ:

$$R^2=\frac{a^2\left(1-{\epsilon }^2\right)}{\left(1-{\epsilon }^2{\mathrm{Cos}}^2\varphi \right)}$$

(5)

As Kepler deduced, for astronomical applications, the position m of an orbiting object in its elliptical path is best expressed in terms of the (r,θ) coordinates centered on the real focus, rather than in either the Cartesian or the polar coordinate systems centered on the ellipse (Figure 1). The link between these coordinates is found by applying the law of cosines and Equation (2) to the triangle whose apices are the two foci and point m, which yields [17] the radial distance r as a function of θ:

$$r=\frac{a\left(1-{\epsilon }^2\right)}{\left(1+\epsilon \mathrm{Cos}\theta \right)}$$

(6)

3.2. The Image Coordinate System for the Ellipse and Links between Coordinate Systems

Alternative coordinate systems are of long-standing interest [9,10,18], yet few authors [7,11,12,19] have recently explored the (r_i, α) coordinate system centered on the image focus. Nevertheless, useful geometric relationships and new insights are revealed by considering the image focus, as follows:

Because of symmetry (Figure 1), an analogue of Equation (6) exists in the (r_i, α) coordinates [11]:

$$r_i=\frac{a\left(1-{\epsilon }^2\right)}{\left(1-\epsilon \mathrm{Cos}\alpha \right)}$$

(7)

Trigonometric identities can be used to derive several useful relationships for the relations between position angles. For example [11]:

$$\left(tan\frac{\alpha }{2}\right)=\frac{1-\epsilon }{1+\epsilon } \left(tan\frac{\theta }{2}\right)$$

(8)

By differentiating Equations (2), (6), and (7), we find that:

$$r_{i}d\alpha =rd\theta$$

(9)

This important relation can also be reasoned from the symmetrical positions of the foci. It has many uses; for example, an alternate derivation of Equation (8) uses Equations (6) and (7) to express the radii in terms of their complementary angles, and then integrating each side.

We also derived a remarkably simple relationship between the Cartesian coordinate system and the two polar coordinate systems centered on the foci:

$$r=a-\epsilon x ; r_i=a+\epsilon x.$$

(10)

Equation (2) follows by simply adding both parts of Equation (10). Note that r = r_i when x is zero (cf. Figure 1). Moreover, the radius of a circle is simply a, because the ellipticity of a circle is zero, and r_i and r have merged.

Equation (10) is easy to prove with reference to Figure 1. Consider the two right triangles whose bases lie along the x axis, and whose hypotenuses are r_i and r. The vertical height H of these triangles below point m is identical. Using Pythagoras’ theorem for each triangle gives:

$$H^2=r^2-{\left(x-a\epsilon \right)}^2=r_i^2-{\left(x+a\epsilon \right)}^2.$$

(11)

Introducing Equation (2) into the right-hand side (rhs) of Equation (11) and collecting terms yields Equation (10). Additional properties of the (r_i, α) coordinate system are provided below, including a new geometrical constant related to the product of the two radii.

3.3. Reduction of Elliptical Geometry to 1-Dimension

Importantly, Equation (10) shows that each of the radii r and r_i depend linearly on the x-coordinate, and are independent of the y coordinate and of any angle. Thus, in a polar coordinate system centered on either focus, the two-dimensional elliptical path of an orbiting body can be described in terms of its linear position along the x-axis alone, thus reducing the two-dimensional problem to a single dimension. This facet underlies Kepler’s remarkable insight to describe the orbital dynamics of a planet in terms of its relationship to the Sun, which occupies a focus, as codified by his first law. Uni-dimensionality of the orbiter’s position also explains why the kinetic energy, potential energy, and velocity of the orbiter, or Newton’s gravitational force, depend only on the radial distance between the two bodies.

4. Angular Relationships and Links to Momentum and Energy

4.1. Reflection Properties Define the Key Angular Variable

The reflection property of the ellipse requires that a ray emanating from one focus will be reflected to the other, as shown in the expanded view of the ellipse (Figure 2). The triangle whose vertices are the orbiting point m and the two foci, F and F_i, has a vertex angle denoted as 2δ. This construction is useful because the angle between the tangent to the ellipse and the perpendicular to r_i (dashed line), which in the limit is parallel to the arc length r_idα, is also equal to δ, from inspection of Figure 2. Similar reasoning shows that the angle between the tangent to the ellipse and the perpendicular to radius r (not shown), which is parallel to arc rdθ, is also equal to δ.

Because the sum of the internal angles of a triangle is 180 degrees:

θ = α + 2δ

(12)

Finally, applying the law of cosines to this triangle, while now using its base as the hypotenuse, gives:

$${\left(2a\epsilon \right)}^2=r^2+r_i^2-2r{r}_i\mathrm{Cos}\left(2\delta \right)$$

(13)

The following trigonometric identity holds for any angle:

$$\mathrm{Cos}\left(2\delta \right)=2{\mathrm{Cos}}^2\delta -1$$

(14)

Replacing Cos(2δ) by the expression on the right-hand side of Equation (13), substituting 2a−r for r_i, and algebraic manipulation yields the following useful relationship, and two alternatives:

$$\mathrm{Cos}\left(\delta \right)=\frac{a\sqrt{1-{\epsilon }^2}}{\sqrt{r\left(2a-r\right)}}=\frac{a\sqrt{1-{\epsilon }^2}}{\sqrt{r_i\left(2a-r_i\right)}}=\frac{a\sqrt{1-{\epsilon }^2}}{\sqrt{r{r}_i}}$$

(15)

It directly follows that the following product has a constant value at any point along an ellipse:

$$r{r}_i{\mathrm{Cos}}^2\delta =a^2\left(1-{\epsilon }^2\right)$$

(16)

Equation (16) defines a new, fundamental property of an ellipse that we term the “radial component product”, and was proven by geometry alone, but it must also apply to a dynamic orbiter. This product provides a useful companion to the familiar radius sum, Equation (2).

4.2. Relationship between Angular Momentum and the Apex Angle

The radial component product (16), though purely geometrical and general, is related to several dynamical constants for the special case of orbits. First, the angular momentum L of an orbiting mass m is known to be conserved, and specifically is equal to m times the cross product of the radius r and the tangential velocity v. Thus:

$$\overrightarrow{L}=-m\overrightarrow{v}\times \overrightarrow{r}, \mathrm{so} L=mvr\mathrm{Cos}\left(\delta \right)$$

(17)

Knowledge of Cosδ defines the scalar magnitude of angular momentum, which greatly simplifies the analysis below. Introducing L into Equation (16), and rearranging, shows that the following quantity on the left, which has units of time, is another orbital constant:

$$\frac{r_{i}Cos\delta }{v}=\frac{a^2\left(1-{\epsilon }^2\right)m}{L}=\frac{T\sqrt{1-{\epsilon }^2}}{2\pi }$$

(18)

Equation (18) shows that this new constant can be expressed in terms of the angular momentum or, as shown below, is directly proportional to the orbital period T. Many relationships exist between dynamic variables, due to conservation laws and the Virial theorem, as follows.

4.3. Restrictions Imposed by Conservation Laws

The total energy TE of an orbiting object is the sum of its kinetic energy (KE) and its potential energy (PE), and is constant throughout the motion:

TE = PE + KE

(19)

where the kinetic energy is mv²/2, where v is tangential velocity. The potential energy of mass m as a function of its radial distance r from another mass M is found by integrating Newton’s gravitational force law between that position and infinity.

$$PE={{\displaystyle \int }}_r^{\infty }Fdr={{\displaystyle \int }}_r^{\infty }\frac{GMm}{r^2}dr=-\frac{GMm}{r}$$

(20)

where G is the gravitational constant. Note that PE is negative, as mass m has “fallen” into the energy well of mass M.

A compact value for the TE of an orbit can be derived by comparing relationships at the apsides, represented by points A and P in Figure 1, and recognizing that conservation laws require that TE and L are constants at all orbital positions. Points A and P are convenient because Cos(δ) is unity at both locations, so Equation (17) provides:

$$L=m{v}_A a\left(1+\epsilon \right)=m{v}_P a\left(1-\epsilon \right)$$

(21)

where the appropriate values for r_A and r_P have been incorporated (see Figure 1), and v_A and v_P are the tangential velocities at these points. Note that v_A can be expressed in terms of v_P:

$$v_A=v_P \left(1-\epsilon \right)/\left(1+\epsilon \right)$$

(22)

Because the total energies at A and P are identical, Equation (19) provides:

$$TE=\frac{-GMm}{a\left(1+\epsilon \right)}+\frac{m{v}_A^2}{2}=\frac{-GMm}{a\left(1-\epsilon \right)}+\frac{m{v}_P^2}{2}$$

(23)

Introducing Equation (22) into Equation (23), plus algebraic manipulation, allows determination of v_P:

$$v_P^2=\frac{GM}{a}\frac{1+\epsilon }{1-\epsilon }$$

(24)

The total energy at P can be computed by introducing this result into the rhs of Equation (23). This yields the total energy of the orbiter at point P, but the conservation of energy ensures that this result must apply to all orbital points. Thus:

$$TE=\frac{-GMm}{2a}$$

(25)

Now enough is known to compute the tangential velocity at any point. Again, using Equation (19):

$$-\frac{GMm}{2a}=-\frac{GMm}{r}+\frac{m{v}^2}{2}$$

(26)

Minor rearrangement provides the tangential velocity:

$$v=\sqrt{GM\left(\frac{2}{r}-\frac{1}{a}\right)}$$

(27)

a result known as the vis viva equation [20]. For completeness and convenience,

Table 1. Coordinates and Velocity (Equation (27)) for Special Orbital Positions.

Point	r	θ	α	ri	Cosδ	$\mathit{v}$	$\frac{{\mathit{r}}_{\mathit{i}}\mathit{C}\mathit{o}\mathit{s}\mathit{\delta }}{\mathit{v}}$
P periapsis	$a\left(1-\epsilon \right)$	0	0	$a\left(1+\epsilon \right)$	1	$\sqrt{\frac{GM}{a}}\sqrt{\frac{1+\epsilon }{1-\epsilon }}$	$\sqrt{\frac{a{c}^2}{GM}}$
A apoapsis	$a\left(1+\epsilon \right)$	0	0	$a\left(1-\epsilon \right)$	1	$\sqrt{\frac{GM}{a}}\sqrt{\frac{1-\epsilon }{1+\epsilon }}$	$\sqrt{\frac{a{c}^2}{GM}}$
C above center	$a$	π-ArcCos[ε]	ArcCos[ε]	$a$	$\sqrt{1-{\epsilon }^2}$	$\sqrt{\frac{GM}{a}}$	$\sqrt{\frac{a{c}^2}{GM}}$
B latus rectum	$a\left(1-{\epsilon }^2\right)$	π/2	ArcTan$\left[\frac{1-{\epsilon }^2}{2\epsilon }\right]$	$a\left(1+{\epsilon }^2\right)$	$\frac{1}{\sqrt{1+{\epsilon }^2}}$	$\sqrt{\frac{GM}{a}}\sqrt{\frac{1+{\epsilon }^2}{1-{\epsilon }^2}}$	$\sqrt{\frac{a{c}^2}{GM}}$
D latus rectum	$a\left(1+{\epsilon }^2\right)$	π-ArcTan$\left[\frac{1-{\epsilon }^2}{2\epsilon }\right]$	π/2	$a\left(1-{\epsilon }^2\right)$	$\frac{1}{\sqrt{1+{\epsilon }^2}}$	$\sqrt{\frac{GM}{a}}\sqrt{\frac{1-{\epsilon }^2}{1+{\epsilon }^2}}$	$\sqrt{\frac{a{c}^2}{GM}}$

summarizes velocities and geometrical properties at the special points. Note that r_iCos(δ)/v is constant for all (right column), as predicted by Equation (18).

Equation (27) can be used to determine the angular momentum L, defined in Equation (17). The value for L is easily computed at either apside, where Cos(δ) is unity, but again the resultant value must apply to any orbital position because L is constant. Thus, for any particular orbit, L is:

$$L=m \sqrt{GMa }\sqrt{1-{\epsilon }^2}$$

(28)

4.4. An additional Restriction from the Virial Theorem

The Virial Theorem of Clausius applies to bound systems with cyclical motions and space and time symmetry [21]. The Virial Theorem is a direct consequence of a mathematical identity known as the Lagrange-Jacobi relationship, and for a bound system, yields a direct relationship between the average KE and the average PE. For an inverse square force law, the coefficient relating these two energies is −2, and thus:

$$\overline{PE}=-2\overline{KE}$$

(29)

where the bars denote time averages over the cycle. Note that this relationship is independent of conservation of energy (Equation (19)). The combined constraint for an orbiting system is:

$$TE=\overline{PE}+\overline{KE}=-\overline{KE}=\overline{PE} /2$$

(30)

4.5. Many Conservation Laws and Constants Describe Elliptical Orbits

Interestingly, it can be shown that $\overline{PE}$ and $\overline{KE}$ are equal to the instantaneous values for PE and KE at point C (Figure 1), where r and r_i are identical and equal to a. It is also easy to show from Equation (10) that the spatial average of r is also equal to a. That the instantaneous values of these quantities at point C are equal to their various averages may come as a surprise, but in fact is a consequence of the dynamics of this highly symmetrical system.

In short, many quantities are constant during an elliptical orbit (

Table 2. Dynamic and Geometric Constants for any Position along an Elliptical Orbit.

Name	Symbol	Constant Value	Alternate Form of the Constant Value	Single Point Representation
Total Energy	TE	$-\frac{GMm}{2a}$
Angular Momentum	L	$m\sqrt{GMa\left(1-{\epsilon }^2\right)}$		$mvr\mathrm{Cos}\delta$
Orbital Period	T	$2\pi \sqrt{\frac{a^3}{GM}}$		$\frac{2\pi r_i\mathrm{Cos}\delta }{v\sqrt{1-{\epsilon }^2}}$
Radius Sum	r + ri	$2a$		r + ri
Radial Component Product	$r{r}_i{\mathrm{Cos}}^2\delta$	$a^2\left(1-{\epsilon }^2\right)$	$c^2$	$r{r}_i{\mathrm{Cos}}^2\delta$
Time Constant	$\tau$	$\sqrt{\frac{a{c}^2}{GM}}$	$\frac{T \sqrt{1-{\epsilon }^2}}{2\pi }$	$r_i\mathrm{Cos}\delta /v$

). While the first four tabulated quantities are well known, the latter two are new. The “radius sum” and the “radial component product” are purely geometrical properties of the ellipse, because they are defined and proved using geometry alone. Yet, the constant value for the latter geometric property is equal to the product of two dynamic constants, the angular momentum divided by m, and the new “time constant.” These new geometrical properties of the ellipse, in particular Equations (15) and (16) that define the angle δ, provide insight into angular momentum and simplify the calculation of orbital progress, as shown below.

5. Exact Orbital Period and Elapsed Orbital Time

5.1. Circular Orbit

Kepler’s third law states that the square of the orbital period T is proportional to the cube of the “distance to the Sun.” Confirmation for a circular orbit is provided by equating Newton’s gravitational force with the centripetal force:

$$\frac{GMm}{R^2}=m{\omega }^{2}R ; T^2=\left(\frac{4{\pi }^2}{GM}\right)R^3$$

(31)

Here, ω is the angular velocity, which is constant for a circular orbit, so in this case, it must equal 2π/T, providing the equation on the right.

5.2. Spatial vs. Temporal Averages

Matters are less obvious for an elliptical orbit. Most but not all textbooks, and many analytical derivations of the third law [17,20], indicate that Kepler’s “mean distance” refers to the semimajor axis a. Kepler’s [16] own statements indicate that his “mean distance” is equal to the radial distance of the orbiter at point C of Figure 1, which indeed is equal to a. Although the result is somewhat surprising, it is easy to integrate Equation (10) to prove that the mean value of r, when averaged over distance x, is also equal to a, yet this cannot be the mean radial distance when averaged over time, because the velocity of the orbiter is the highest when close to the Sun, and the lowest when far away. Interestingly, the mean value of r, when averaged over angle θ (or over α), is instead equal to c. Therefore, Kepler’s use of a spatial average for the mean distance requires examination, because he mostly discusses the means of “daily arcs,” that must represent averages of Brahe’s daily measurements, which are time averages.

The time required to move an incremental distance along an elliptical arc at velocity v is:

$$dt=\frac{rd\theta }{v\mathrm{Cos}\delta }=\frac{r^{2}d\theta }{\sqrt{GMa\left(1-{\epsilon }^2\right)}}$$

(32)

where the Cosδ term corrects for the angular difference between the radial arc rdθ and the elliptical path, and the equality on the right follows from substituting the relations for velocity and Cosδ (Equations (15) and (27)). Notably, the numerator on the right is equal to twice the area of the arcuate elliptical segment anchored by r to the real focus, yielding:

$$dt=\frac{2dA}{\sqrt{GMa\left(1-{\epsilon }^2\right)}} or dt=\frac{2m}{L} dA$$

(33)

Finally, by using Equation (9), Equation (32) can be recast as the important pair:

$$\frac{d\theta }{dt}=\frac{L}{2m{r}^2} and \frac{d\alpha }{dt}=\frac{L}{2mr{r}_i}$$

(34)

Our result shows that equation A26 of [11] is missing a factor of 2m, so the earlier representation is numerically and dimensionally incorrect.

5.3. Confirmation of Kepler’s 2nd and 3rd Laws in Two, New Approaches

The direct proportionality between dt and dA in Equation (33) confirms Kepler’s second law. Furthermore, his “equal areas in equal times” law is far more general than Kepler realized. The expression on the rhs of Equation (33) can also be derived (e.g., [17]) solely from the vector definition of angular momentum for the path of an object governed by any central force law, so the object’s path need not be elliptical, nor the force affecting it an inverse square.

The time required to complete a complete orbital circuit can now be determined by integrating Equation (32) from 0 to 2π, after expressing r in terms of θ using Equation (6). Ambiguities concerning the trigonometric quadrants can be avoided by integrating from zero to π, and doubling the result. Thus:

$$T=2{{\displaystyle \int }}_0^{\pi }dt=\frac{2a^{\frac{3}{2}} {\left(1-{\epsilon }^2\right)}^{\frac{3}{2}} }{\sqrt{GM}}{{\displaystyle \int }}_0^{\pi }\frac{d\theta }{{\left(1+\epsilon \mathrm{Cos}\theta \right)}^2}=\frac{2\pi a^{\frac{3}{2}}}{\sqrt{GM}}$$

(35)

The rhs of Equation (35) utilizes Gradsheteyn and Ryzhik’s [22] solution to the definite integral. This procedure confirms Kepler’s third law in a new way. Equation (35) confirms that the proportionality constant is the same as that for a circular orbit, given in Equation (31), and proves that the “mean distance” required for this problem is, indeed, the semi-major axis a.

5.4. Instantaneous Values and Quantification of Orbital Progression

The method applied above can be used to determine the time associated with any orbital position. This is done by replacing T by t of the left-hand side (lhs) of Equation (35), which requires changing the definite integral to an indefinite integral. Its solution [22] allows Equation (35) to be recast as:

$$t=\frac{T}{2\pi }\left\{2\mathrm{ArcTan}\left[\sqrt{\frac{1-\epsilon }{1+\epsilon }} \mathrm{Tan}\left[\frac{\theta }{2}\right] \right]-\frac{\epsilon \sqrt{1-{\epsilon }^2} \mathrm{Sin}\left[\theta \right]}{1+\epsilon \mathrm{Cos}\left[\theta \right]}\right\} for \pi \ge \theta \ge 0$$

(36a)

$$t=\frac{T}{2\pi }\left\{2\pi -2\mathrm{ArcTan}\left[\sqrt{\frac{1-\epsilon }{1+\epsilon }} \mathrm{Tan}\left[\frac{\theta }{2}\right] \right]+\frac{\epsilon \sqrt{1-{\epsilon }^2} \mathrm{Sin}\left[\theta \right]}{1+\epsilon \mathrm{Cos}\left[\theta \right]}\right\} for 2\pi \ge \theta \ge \pi$$

(36b)

Integration of Equation (34) provides a similar analytical result for orbital progress in terms of angle α about the image focus:

$$t=\frac{T}{2\pi }\left\{2\mathrm{ArcTan}\left[\sqrt{\frac{1+\epsilon }{1-\epsilon }} \mathrm{Tan}\left[\frac{\alpha }{2}\right] \right]-\frac{\epsilon \sqrt{1-{\epsilon }^2} \mathrm{Sin}\left[\alpha \right]}{1-\epsilon \mathrm{Cos}\left[\alpha \right]}\right\} for \pi \ge \alpha \ge 0$$

(37a)

$$t=\frac{T}{2\pi }\left\{2\pi -2\mathrm{ArcTan}\left[\sqrt{\frac{1+\epsilon }{1-\epsilon }} \mathrm{Tan}\left[\frac{\alpha }{2}\right] \right]+\frac{\epsilon \sqrt{1-{\epsilon }^2} \mathrm{Sin}\left[\alpha \right]}{1-\epsilon \mathrm{Cos}\left[\alpha \right]}\right\} for 2\pi \ge \alpha \ge \pi$$

(37b)

Although these expressions are more complicated than others in this report, they are reasonably compact, and provide the exact orbital time since perihelion, without iteration. It is also convenient that this pair applies to the forward and reverse paths, respectively. None of the textbooks (e.g., [13,14]) or websites that we examined provide a direct analytical relation between position and time for any ellipticity. Only the exact differential relationships, or series approximations whose accuracy depends on ellipticity, are available along the entire elliptical path.

Finally, it can be shown by integration, or far more simply by introducing Equation (25) into Equation (30), that the mean potential energy, and hence, the mean kinetic energy of the orbiter, when averaged over time, are both equal to their instantaneous values at point C. These time-averaged energies are the essential inputs for the Viral Theorem.

In short, the instantaneous values for the orbiter at point C are equal to its mean radial distance from either focus when averaged over space, but not when averaged over time, and provide the mean kinetic and potential energy of the orbiter when averaged over time, but not over space.

6. Orbital Clock Approximation

Many historical works have attempted to define a point of observation around which a planet would move in a uniform manner. Symmetry requires that any such point must lie along the line of apsides. The following analysis confirms that the image focus serves well as this location [7]. However, no point can provide a perfect solution to this problem, as follows:

The image focus provides an approximation to this problem, while simultaneously providing a perfect answer at the apsides and at two other points. Were observations made from any other point along the semimajor axis, it follows that the apparent angular velocities at the aphelion and perihelion cannot be identical. Thus, no point of observation exists from which the orbiter would exhibit perfectly uniform motion.

Equation (18) about the image focus has interesting affinities with Equation (32) about the real focus. Both have units of time, and when the left-hand side of Equation (18) is multiplied by da, the result represents an incremental arc length divided by velocity, i.e., an incremental time dτ. Specifically:

$$d\tau =\frac{r_i\mathrm{Cos}\delta }{v}d\alpha =\frac{T\sqrt{1-{\epsilon }^2}}{2\pi }d\alpha =\frac{\sqrt{1-{\epsilon }^2}}{\omega }d\alpha$$

(38)

where T/2π has been written in terms of the average angular speed ω. Because the multipliers of dα are orbital constants, the time increment dτ is directly proportional to dα. This incremental time index would be a perfect, uniform measure of orbital progress, equal to dt, if the Cosδ term was located in the denominator instead of the numerator, so that the arc length under consideration would be perfectly congruent with the perimeter of the ellipse. Nevertheless, angle δ is small for orbits of small eccentricities, so this defect is small. It follows that, if ε is small, then α~ωt. This linear relation is exact at several orbital points, and the maximum angular deviation between α and ωt is less than ε² for eccentricities smaller than 0.9. For small ε, the maximum deviation is ε²/4. Thus, using this excellent linear approximation, Equation (7) becomes:

$$r_i\cong \frac{a\left(1-{\epsilon }^2\right)}{\left(1-\epsilon \mathrm{Cos}\left[\omega t\right]\right)} \mathrm{or} r \cong 2a-\frac{a\left(1-{\epsilon }^2\right)}{\left(1-\epsilon \mathrm{Cos}\left[\omega t\right]\right)}$$

(39)

Although approximate, Equation (39) provides a relationship of a type that has heretofore been unavailable, which is a compact, direct link between the two major variables of orbital dynamics, namely radial position and time. Note that the other equations in this paper only describe the spatial coordinates of the ellipse, or relate velocity, energy, etc. to its placement along the ellipse, but provide no information about when the object arrived at any position.

The useful approximation of Equation (39) also provides a very helpful visualization: an orbital clock lies hidden in the ellipse. The radial hand r_i of this clock rotates at nearly steady angular speed ω around the image focus. In detail, this clock runs slightly fast during part of its path, and then runs slightly slow, but always returns the exact time at the apsides. Figure 3 illustrates the utility of this clock, which is far easier to visualize than to conceptually divide an ellipse into irregular, arcuate segments having equal areas.

7. Validation against Solar System Observations

Figure 3b illustrates the accuracy of our approximation for c/a = 0.9, corresponding to an eccentricity of ~0.43589. The 8 planets have significantly lower ε (see below). The 199 moons listed in the NASA fact sheets [23] mostly have significantly lower eccentricity, such that only 24 moons have ε > 0.436. Most of the latter are small, captured bodies with highly tilted and/or retrograde orbits. The only regular satellite with high ε is Nereid, the 3rd largest moon of Neptune. Thus, the accuracy shown for the orbital clock in Figure 3b holds for all planets and all but one of the significant moons in the solar system.

Regarding the planets, we explore two with contrasting orbital characteristics. Mercury is considered because its orbital eccentricity of 0.2056 (about ½ of that shown in Figure 1) is higher than that of all other planets and so is the planetary “worst case.” Of the moons [23], 84 have lower ε: these include almost all the regular moons. Mercury is the innermost planet, and is of particular interest as a test of relativity [5]. Jupiter is considered because it is the most massive planet and has a lower eccentricity of 0.0487 (about 1/9th of that shown in Figure 1), and is located outboard of Earth’s orbital radius (Figure 4).

Although Neptune would be of interest, being the outermost planet, its orbit is nearly circular and modern observations are few due to its ~160-year period. Saturn and Uranus have similar properties as Jupiter (Figure 4), so the results will be similar. Mars is not considered here because it is perturbed by Jupiter [4,6]. Earth is not considered because it provides the geocentric reference point. Venus’ orbit is nearly circular. Thus, calculations for Mercury and Jupiter suffice.

7.1. Mercury

The accuracy of the Equation (39) approximation is illustrated by a position-time calculation for Mercury (Figure 5), the planetary “worst case.” All that is required for this orbital calculation is Equation (39), and data for the aphelion (A = 69.818 Mkm), perihelion (P = 46.000 Mkm), and sidereal orbital period T (87.969 days) [23]. These values suffice because 2a = A + P, ε = (A − P)/(A + P), and ω = 2π/T. Of course, to index the calculation to terrestrial time, the position of Mercury on any specific date, most conveniently the calendar time of any perihelion or aphelion of its orbit, is needed.

The slight discrepancy of our calculation in Figure 5 from accurate ephemeris data [24] stems from more factors than Equation (39) being an approximation. The most important factor is that Earth’s non-planar orbit affects measurements of Mercury against the stars or with respect to the Sun.

In detail, although the barycenter of the Earth and Moon moves in a nearly circular orbit around the Sun, the geocenter does not. The orbit of the geocenter is non-planar and varies over the month as a consequence of the Moon’s existence and relatively large mass. Hence, the ecliptic plane, which is an average of the Earth’s position over the year, is taken as the reference point (e.g., [25]). Due to the existence of the Moon, Earth’s geocenter is rarely on the elliptic plane, so the accuracy of geocentric observations of Mercury are affected.

Figure 6 illustrates the accuracy of the orbital clock approximation in another manner. The angular speed of Mercury’s heliocentric orbit varies more than twofold, and its geocentric orbit varies from prograde to retrograde, yet the angular velocity about the image focus is nearly constant, with a variation shown of about ±2%.

7.2. Jupiter

Because accuracy was established for Mercury, the worst case among the planets, the utility of our approximation is explored for Jupiter. Applying our formulation to Jupiter shows that relationships among the various coordinate systems are similar for outer and inner planets. Jupiter’s eccentricity (0.0487) is much lower than that of Mercury, but its aphelion (A = 816.363 Mkm), perihelion (P = 740.595 Mkm), and sidereal orbital period T (4332.6 days) are much greater [23].

Figure 7 shows Jupiter’s orbit as observed from the geocenter [24], the Sun’s body center [24], and from the image focus for the period 1980 to December 2022, all projected to the xy plane. The “spirograph” path of Jupiter as observed from Earth is the most complicated, with Jupiter exhibiting apparent retrograde motions about 30% of the time. These retrograde segments occur when Jupiter is the closest to Earth, that is, during times of opposition. In contrast, Jupiter follows an almost perfect elliptical orbit in either the heliocentric or image reference frames, with the latter simply being a rotated version of the former.

It follows that the angular velocity of Jupiter as seen from Earth is highly variable, varying from +0.2406 deg d⁻¹ (prograde) to –0.132 deg d⁻¹ (retrograde). In contrast, the heliocentric angular velocity is monotonic and varies only by about 10% from its mean value, i.e., from 0.0753 to 0.0917 deg d⁻¹. However, the angular rate about the image focus varies by only ±½ % from its mean value, with a total range of 0.0829 to 0.0837 deg d⁻¹.

8. Discussion

8.1. Pedagogy

Our analysis only uses trigonometry, symmetry, and basic calculus, and so should be tractable to undergraduate students who have little or no familiarity with either vector methods or elliptical functions. Many of the proofs and examples, including the new approximation for Mercury’s orbit, can be converted into student exercises.

The familiar advice to “chose the right coordinate system” to analyze a particular physical problem is important. Kepler showed that analyzing planetary orbits as elliptical paths about a focus occupied by the Sun reveals its most essential secrets. A useful corollary is that it can also be helpful to analyze a system from as many different viewpoints and perspectives as possible. As the Greeks taught us, once a geometric truth is revealed, matters that appear complex become simple.

Discerning whether a possibility is reasonable often involves visualization. Figure 6 and Figure 7 illustrate the pitfalls of geocentrism, while suggesting the historical motivation for invoking epicycles. Another approach illustrated in this paper is to consider specific situations and limiting cases: here, the ideal elliptical orbit. Combining elliptical and hyperbolic orbits in a single equation (e.g., [17]) is confusing to students, and short shrifts the key feature of periodicity in stable orbits and its historical importance in defining time.

Training students in physics and related fields begins with equations that illustrate the basic principles of physics. Part of this process involves validation, through comparison to measurements, as provided in Section 7. The fundamental equations of physics are simple, commensurate with the parsimony that leads to truth. This principle has been overshadowed in modern research, due to the expected progression towards solving increasingly difficult problems. Specifically, solutions are commonly presented numerically, making them quite difficult to evaluate. Numerous examples exist. Solutions are also presented in terms of very complicated mathematics. One example is use of an equation with >100 integrals to represent the shape of a spinning, inhomogeneous body [26], when a simple analytical function suffices [27]. Our analytical formulae for orbital time, Equation (37) and its simple approximation (39) are other examples. Perhaps the analytical approach presented here, which solves a centuries old problem of whether a location to observe uniform planetary motion exists [7,8], could motivate undergraduate students to further their mathematical skills, and encourage those further along the educational process to place greater emphasis on parsimony.

8.2. Stability and Evolution of Keplerian Elliptical Orbits

To place our results in context, we summarize previous work. Circular orbits are expected for one tiny mass orbiting a very large mass. This case resembles our Solar System, as even Jupiter’s mass is tiny (~0.3%) compared to that of the Sun, and eccentricities are small (Figure 4). Following Newton and Gauss, the perturbing forces of the various planets on any given planet have been modelled in terms of planets interacting as pairs, and these effects were then summed (e.g., [4]). Otherwise, planetary orbits would have been so complex that Kepler might never have discovered his famous laws.

Under the assumption that elliptical orbits are the stable configuration, almost all efforts to quantify interplanetary perturbations have viewed orbital precession as the consequence. However, dissipative forces are absent so energy is conserved (illustrated by Equation (25)), under which condition, eccentricities, and inclinations of the orbits can change. On this basis, a previous study [6] deduced that interplanetary interactions are elongating and tilting the orbits. Figure 4 shows that the inner planets are being pulled towards the orbital plane of Jupiter, whereas the outer planets appear to be converging to a plane with an inclination between those of Jupiter and Saturn. The strong influence of massive Jupiter is also evident in the orbital eccentricity of the planets, which is the largest for tiny Mercury and also for small Mars adjacent to Jupiter. Importantly, Mercury, which is pulled outward by all planets, is the most eccentric, whereas Neptune, which is the least affected by perturbations due to its distance and long orbital period and is only pulled inwards, has one of the lowest eccentricities.

This comparison suggests that planetary orbits were perfectly circular when the Solar System formed [6]. Although energy is conserved during evolution, angular momentum is not conserved because L depends on ε (Equation (28)), and because orbits of both planets of a pair are elongated during their interaction. This was not recognized in a previous study [6]. Cancellation of the imbalanced forces and torques over the orbit in these 3-body interactions, except at the close and far passes, makes interplanetary perturbations a small and periodic perturbation. Evolution is slow, even for Mercury (Figure 4). Findings in the present paper confirm that the modern elliptical orbits of the planets are not original. Circular orbits must be the starting point, as they have the endmember value of ε = 0.

Given that elliptical orbits result from solar system evolution, the image clock (Figure 3) is necessarily an approximation. Uniform circular motion existed only when the Solar System formed, in alignment with philosophical and theological precepts (e.g., [8]).

8.3. Utility of Our Exact and Approximate Dynamical Equations

Orbits of the planets and moons must be modeled because observations are made from Earth, which has a non-planar orbit, and because the lunar and Julian years are not identical. Projections of planet orbits forward and backward in time are accurate only over a restricted interval (e.g., [25]). Tradeoffs exist between the span of time and the accuracy of the 3-dimensional positions of the planets and moons. Updates are now made roughly every 30 years.

Our equations are simpler than the 9 term polynomials currently used to describe planetary positions (e.g., [25]). From a starting point, typically the traditional but evolving spring equinox, time can computationally be run backwards and forwards.

The exact Equations (36) and (37) and the approximation of Equation (39) in this report can potentially be used to quantify and improve understanding of precession, which is currently unconstrained for planets and moons (see [6,28]), and the complex lunar and terrestrial orbits. Regarding long-term solar system evolution, utilizing these equations may be the optimal approach.

9. Conclusions

Geometry lies at the heart of many advances in art, architecture, and physical science. These include fundamental insights into the nature of the atom; the structure of molecules, DNA, and crystals; the shape of planets and stars; and Kepler’s discovery regarding the elliptical nature of solar system orbits.

This paper offers a new perspective on orbits based on a geometrical analysis of the polar (r_i, α) coordinate system centered on the image focus. Considering symmetry of the ellipse, particularly its reflection property, disclosed a new geometric constant that quantifies the angle δ between radial arcs and the perimeter of the ellipse. Knowledge of this angle greatly simplifies dynamic calculations and lead to a new and convenient equation that provides the exact orbital time associated with any orbital position. Moreover, the angular velocity of an orbiting object is nearly constant about the image focus, in great contrast to its angular velocity as viewed from any other point. This property provides an orbital clock visualization that has good accuracy for orbits having low eccentricity, i.e., the solar system planets and most of their moons. Our results, combined with earlier work, show that orbits of the solar system were originally circular.