A straightforward method of determining the eccentricity of Earth's orbit and the position of aphelion and perihelion relative to the vernal equinox from solstice and equinox times is described. The only assumption made is that the orbital eccentricity is small. Using dates for these phenomena adopted from a desk calendar gives the eccentricity to an accuracy of about 10%.

The eccentricity of Earth's orbit is related to the elapsed time between successive equinoxes and solstices; the time required to travel from northern hemisphere autumnal equinox (AE) through winter solstice (WS) to vernal equinox (VE), about 179 days, is less than that required to go from vernal equinox through summer solstice (SS) and back to the autumnal equinox, about 186 days. The difference corresponds to only about 2% of the orbital period, so the eccentricity must be slight; also, we can infer that perihelion must occur during the northern winter months, a fact known to the ancient Greeks.

Several papers have presented various approaches to determine the eccentricity of Earth's orbit from solstice/equinox timing data; there are also some textbook treatments. A 1989 paper by Snyder summarizes several older treatments and presents a model where the orbit is treated as a Ptolemaic-like offset circle of varying speed.^{1} However, as Snyder and others point out, these approaches all suffer from limitations such as assuming that the orbit is circular or that perihelion and winter solstice coincide; some involve awkward transcendental functions or series expansions.^{2,3}

An ingenious variant of the circle method was developed by Oostra, who has the Earth in a circular orbit about the origin. The radius of the orbit can be taken to be unity, and he places the Sun offset from the origin at coordinates (*x*, *y*), where *x* and *y* are presumed to be small compared to the radius. Lines parallel to the *x* and *y* axes are projected from the Sun until they intersect the orbit; this divides the circle into four sectors whose areas are greater or less than $\pi /4$ according as the areas of thin rectangles dictated by the values of *x* and *y*. The intersection points of these lines with the circle are taken to define the positions of solstices and equinoxes. By Kepler's second law, the sector areas must be proportional to the known durations of seasons; a consistent chronological ordering of the sectors with the seasons becomes apparent, and ratios of areas to times for different seasons can be used to establish values for *x* and *y*. The eccentricity is then recovered as the ratio of the offset distance to the radius of the circle.^{4,5}

For students familiar with Kepler's laws, however, the offset-circle method may be confusing, since they know that orbits should be ellipses. To avoid this problem, this note describes an alternate method of estimating the eccentricity of Earth's orbit and the location of aphelion/perihelion relative to the vernal equinox. While the end result is equivalent, it starts with the more familiar assumption that the orbit is a low-eccentricity ellipse.

Of course, planetary orbital data can be looked up in any online almanac, where timings will be quoted to an accuracy of a minute. However, to adopt such data to test the method developed here would be disingenuous: Such almanacs are generated by software already programmed with the eccentricity. To keep to the spirit of imagining an ancient but patient observer who determines the relevant dates by noting equinoxes as when the Sun rises due east and solstices as those of longest and shortest noontime shadows, I use dates rounded to one day, adopted from my desk calendar. It turns out that even with this one can do reasonably well, determining the eccentricity to an accuracy of about 10%. I do give precise data for comparative purposes, but the intent is to simulate an “eyeball” approach. This analysis would likely be nothing new to a Kepler or Newton, but I have not found it described in any readily accessible source.

The situation is sketched (not to scale) in Fig. 1. In the top sketch, the ellipse represents Earth's orbit as viewed from above the North pole of the Sun; the eccentricity is greatly exaggerated. The thin horizontal solid line passing through the Sun is the major axis of the orbit. I put the summer solstice just somewhat past aphelion. In reality, this occurs about two weeks before aphelion, but the diagram is drawn for convenience, and this will not affect the following argument. The apsidal angle $\varphi $ is measured counterclockwise from the major axis with aphelion corresponding to $\varphi =0o$. Ultimately, the goal is to determine the positional angle of vernal equinox, $\varphi VE$, relative to this direction. In effect, this is reversed from the usual astronomical practice of treating the equinox position as “fixed” and determining the location of aphelion or perihelion, but this makes the formulation more straightforward.

A key argument in this analysis is that solstices and equinoxes are separated by $90o$ in apsidal angle. For readers unfamiliar with this notion, it can be argued as follows. Figure 1(b) shows a view in the plane of the ecliptic, perpendicular to the WS-Sun-SS line (*not* perpendicular to the major axis of the ellipse). At summer solstice, the Sun will appear overhead as far north as it ever does. To have the same extreme southern latitude at winter solstice on the left side of the sketch, the summer and winter solstice positions must be $180o$ apart. An identical argument can be made for the positions of vernal equinox and autumnal equinox; they too must be $180o$ from each other.

Now imagine looking toward the Sun from this external vantage point. The Sun can be over the equator only at the moment when Earth is directly in front of or behind it as viewed from this direction. Thus, the SS-WS and VE-AE lines must be perpendicular, hence the $90o$ claim. This is a quite general result, independent of any eccentricity of the orbit.

Orbital dynamics gives us an expression for the time for a planet to travel from apsidal angle $\varphi 1$ to apsidal angle $\varphi 2$ for an orbit of eccentricity *ε*,^{6}

where

and where *T* is the period of the orbit. Additionally, I will use ratios of travel times, so Eq. (2) is not needed.

If *ε* is small, the denominator in the integrand of Eq. (1) can be treated with a first-order binomial expansion to give

With terms of order $\epsilon 2$ and higher neglected, then the *A* of Eq. (2) reduces to $A=T/2\pi $, which renders this solution equivalent to Oostra's offset circle. This is to be expected as both methods are accurate to the first order in *ε*.

Now consider the time to travel from VE to SS. Since $\varphi SS=\varphi VE+90o$, $sin\u2009\varphi SS=cos\u2009\varphi VE$, and we have

Similar expressions for other $90o$ spans can be formulated as

There are two unknowns, $\varphi VE$ and *ε*. Define *τ*_{1} as the ratio of the differences of the elapsed times $(tVE\u2192SS\u2212tAE\u2192WS)$ and $(tWS\u2192VE\u2212tSS\u2192AE)$,

This can be solved for $\varphi VE$,

With $\varphi VE$ in hand, *ε* can be determined. To avoid dealing with powers and roots, it is again helpful to use a ratio to eliminate *A*. Any pair of Eqs. (4)–(7) can be used; I use Eqs. (4) and (6),

where

Table I lists timing data for 2022/2023 phenomena as determined by the United States Naval Observatory; these are included for readers who might wish to undertake more precise calculations. The last column lists the number of elapsed calendar days, not including the day of the “prior event.” These sum to 365, so we can naturally expect some inaccuracy in the results. If you replicate such calculations for other years, beware of subtleties, for example, as measured in Universal Time, the 2022 autumnal equinox occurs in the early hours of September 23, but in the Eastern time zone which my calendar keeps this occurs late in the evening of September 22.

Event . | Date/Time (UT) . | Days since prior event . | Elapsed calendar days . |
---|---|---|---|

Vernal equinox | 20 Mar 2022 15:33 | — | — |

Summer solstice | 21 Jun 2022 09:14 | 92.7368 | 93 |

Autumnal equinox | 23 Sep 2022 01:04 | 93.6597 | 93 |

Winter solstice | 21 Dec 2022 21:48 | 89.8639 | 90 |

Vernal equinox | 20 Mar 2023 21:24 | 88.9833 | 89 |

Event . | Date/Time (UT) . | Days since prior event . | Elapsed calendar days . |
---|---|---|---|

Vernal equinox | 20 Mar 2022 15:33 | — | — |

Summer solstice | 21 Jun 2022 09:14 | 92.7368 | 93 |

Autumnal equinox | 23 Sep 2022 01:04 | 93.6597 | 93 |

Winter solstice | 21 Dec 2022 21:48 | 89.8639 | 90 |

Vernal equinox | 20 Mar 2023 21:24 | 88.9833 | 89 |

Equation (8) gives $\tau 1=\u22123/4$, from which Eq. (9) gives $\varphi VE\u223c262o$; Eq. (10) then gives $\epsilon \u223c0.0152$. (There will be two solutions to Eqs. (8) and (9) separated by $180o$, but one of them will give an unphysical negative eccentricity.) The true eccentricity is $\u223c0.01671$; the estimate is good to $\u223c9%$. The precise USNO numbers give $\varphi VE\u223c256.6o$ and $\epsilon \u223c0.0167$. Perihelion and aphelion occur about two weeks after winter solstice and summer solstice, around January 4 and July 4, respectively. The sensitivity of results to changes in the intervals can be judged by adding a quarter-day in the SS-to-AE calendar interval (when Earth is moving slowly) to account for the true orbital period; this gives $\varphi VE\u223c260o$ and $\epsilon \u223c0.0158$, which halves the error in the latter. Rounding the USNO intervals to the nearest tenth of a day gives $\varphi VE\u223c256o$ and $\epsilon \u223c0.0166$, which essentially eliminates the error. However, this would require patient observation over many years with good timekeeping and calendars.

It is important to point out to students that Earth's orbit and axial rotation are never the beautifully closed, endlessly repeating cycles of textbook drawings; we are constantly subject to gravitational tugs from every body in the solar system. The differential gravitational force of the Sun on the Earth, for example, leads to a precessional motion of the vernal equinox of about 50 s of arc per year. This causes the date of perihelion to advance by about one calendar month every 2000 years, although the effect is not uniform. In Newton's time, perihelia occurred in late December as opposed to the present ∼ January 4. Figure 2 shows the distribution of 400 computed perihelia dates from 1701 to 2100. We have to take the model presented here within the context of these limitations, but it is instructive to see how simple observations served as the foundations of celestial mechanics. A good class exercise might be to have students dig out old calendars and repeat the analysis for various years as a consistency check.

## ACKNOWLEDGMENTS

The author is grateful for the input of three reviewers whose comments resulted in improvements to this paper.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to report.