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Pre-Industrial Elliptic Orbits and Heliocentrism in the Hellenistic Era

Vorpal

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What are the plausibility and long-term effects of elliptic orbits being developed in ancient times?

Regarding the first part, I became interested in whether elliptic orbits could be accessible to the ancients given their observations and conceptual tools. Other than perhaps requiring some implausibly lucky inspirations, it does not seem to be at all impossible for the mathematics and observations of the times to have achieved this. To facilitate this, I will posit two mathematical developments, first a relatively minor one to link ellipses and Greek conception of circular motion as natural, so that they might widen their perspectives to the possibilities, and then a more direct construction of the ellipse linked to Keplerian orbits. The actual point of divergence could occur in multiple ways; ideally, it before the invention of the equant (ca. 150 CE historically), since the following scheme is extremely similar.

To be clear, I don't necessarily mean Keplerian orbits, as the the spatial shape of the orbit does not by itself determine the correct speed of outside a physical theory. Although whether Keplerian orbits can be developed early is also an interesting issue; if elliptical orbits are used, eventual correction to Keplerian is conceptually smaller. Historically, Aristarkhos of Samos was an early proponent of heliocentrism.

...

The first use of epicycles is typically credited to Apollōnios of Perga, ca. 200 BCE, from whom we also get the modern definitions and names for the conic sections (though the names might predate him). Given the predilections of Greek physics of considering circular motion as natural, circles upon circles is a sensible compromise. However, in a direct sense, ellipses are some of the simplest of possible epicycles: a point on a smaller rotating circle that is itself rotating upon a larger one, at the same angular frequency but in oppose directions, traces out an ellipse.

Analytically, z = ½(a+b)exp(iωt) + ½(a-b)exp(-iωt) traces an ellipse of semi-major axis a and semi-minor axis b. If one writes R = a/(1+k) = b/(1-k), the front coefficients become R and Rk, respectively, which is a hypotrochoid formed by a circle of radius R rolling inside a circle of radius 2R with the linked tracing point a distance Rk from the inner circle's centre. Equivalently, one can think of selected points on arbitrary concentric rotating circles, take the vector from the inner rotating point to the outer one, and translate it to the centre. If the circles rotate in the same direction, the result is an epitrochoid, and if in opposite direction, a hypotrochoid.

This could conceivably be discovered very early, e.g. as a slight generalisation of a construction originally developed by Nasir al-Din Tusi (1247). In the Tusi couple, a circle rolls without slipping inside a larger one of twice the size, and a point on the inner circle traces out a line segment, a diameter of the larger. If instead a point inside the rolling circle is picked, the result is an ellipse.

Of course, so far this is just a way to explore the ellipse in a way that conforms to Greek conceptions of natural motion and is not too implausible for them to develop, rather than a good model of an orbit. For an ellipse generated by angular frequency n (= mean motion), semi-major axis a, and eccentricity e, the speed is given by v² = a²n²(1-e²cos²(nt)), which is not even qualitatively right, having a minimum at both apsides.

To try to fix this, one can instead characterise an ellipse and the corresponding orbit in one of two equivalent ways:
(A) Given a line segment AB and points O,F on it such that AO=FB, the locus of points P' such that OP'+FP'=AB is an ellipse with major axis AB and foci O,F.​
The planetary model would have the Sun at F and a uniform rotation about O, which keeps the sum of the distances to the Sun and this shadow focus constant. This is now the common ‘tack-and-string’ definition, though I'm not sure how old the construction actually is; however, that every ellipse has this property is Apollōnios' Conics III.52. The term ‘focus’ (‘hearth’) is anachronistic, being introduced by Kepler in Optics (1604), but Apollōnios did construct those two points for ellipses, calling them ‘the points arising out of the application’, the application essentially being the taking away an appropriate amount from both sides of the major axis.

Another equivalent characterisation of ellipses is the following, which is nicer for eventual correction to Keplerian orbits:
(M) Given a circle with centre O and a radius OP, pick a point F in the interior. Find P' as the intersection of OP and the perpendicular bisector of FP. As P is varied along the circle, the locus of the resulting points P' is an ellipse with foci at O,F and major axis equal to OP.​
Here the trick is to show FP'=PP', so FP'+OP'=OP is constant; the perpendicular bisector of FP is tangent to the ellipse and bisects the angle FP'P, which is related to Conics III.48. Therefore, this construction would not be unnatural to ancient geometers. Under (M), orbit could be thought of as generated from the uniformly rotating circle directly or as being driven by it, with the ellipse as a counter-rotating epicycle as before.
Either way, the model matches Kepler to first order in eccentricity, which for illustrative purposes is particularly easy to check for the angular velocity (of the true anomaly) at the apsides:
(M) θ˙/n = (1±e)/(1∓e) = 1 ± 2e + 2e² + O(e³),
(K) θ˙/n = (1±e)²/√(1-e²)³ = 1 ± 2e + (5/2)e² + O(e³),​
thus having relative error 1-½e²+O(e⁴). Under this model, the maximum and minimum angular speeds are inversions of each other, θ˙₊θ˙₋ = n², while for Kepler, the spatial speeds are instead, v₊²v₋² = (an)². Before the equal-area law is realised, the Tusi couple could perform a finer speed adjustment, which was its original purpose, should observations require it.

The ellipse construction (M) is the chief workhorse a certain simplified geometric proof of Kepler's laws, e.g. in Maxwell's Matter and Motion (1877) and one of Feynman's lectures (1964); Feynman credited it as inspired by Ugo Fano's work on Rutherford scattering. Rather than a uniform rotation in space, as is used here for simplicity and plausibility of what the ancients might have tried first, what's actually going on in Kepler orbits is that hodograph is a circle, i.e. in velocity space, the orbit is a circle with ratio of centre offset to radius equal to e = c/a.

Interestingly, the hodographs of constant specific energy are a pencil of circles discovered by Apollōnios. The elliptic pencil is in in red; the common points of intersection are |v| = an = √(-2ε). The circles in the blue pencil are formed as the locus of points with a constant ratio of distances from the previous two points, as is according to the Apollōnios' characterisation of a circle; they are related to hyperbolic orbits instead and are everywhere orthogonal to the red pencil.
 
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Rufus Shinra

Well-known member
Err, weren't epicycloids used anyway in pre-Kepler astronomy to explain the apparently absurd movements of outer planets, which were going back on their way from the Earth PoV?

https://digitalrepository.unm.edu/cgi/viewcontent.cgi?referer=https://www.google.com/&httpsredir=1&article=1001&context=math_etds
While the models put forth in the Almagest were new in the respect of being presented in a systematic, quantified manner with accompanying rigorous proofs, many of the underlying ideas remained unchanged. Ptolemy believed strongly in the ideas of Aristotle, as passed down from Plato. Ptolemy built his entire system around Aristotle's model in which there is a "fixed earth around which the sphere of the fixed stars rotates every day, this carrying with it the spheres of the sun, moon, and planets."20 All motions were to be circular and uniform, since Ptolemy again considered the circle to be the perfect shape and thus appropriate for modeling the heavens. Indeed, Ptolemy went so far as to separate the fields of physics and mathematics, with the former being applied to earthly, changing things, and the latter to the heavens, which are "eternal and impassible."21

Ptolemy's system uses epicycloids (in the form of deferents and epicycles or the equivalent eccentric circle) to model the motions of the heavenly bodies. In book 3 of the Almagest, Ptolemy set about studying solar motion.
 

Vorpal

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Administrator
Err, weren't epicycloids used anyway in pre-Kepler astronomy to explain the apparently absurd movements of outer planets, which were going back on their way from the Earth PoV?
That seems to use the term in a nonstandard way that to mean epitrochoids instead, or something. An epicycloid is formed by a point on a circle that's rolled without slipping on another circle (ed.). An epitrochoid generalises this to have a linked tracing point an arbitrary distance from the rolling circle's centre; it is the latter than matches (single) epicycles rotating in the same sense. In the Ptolemaic system, there is a point on the rotating circle (deferent) that serves as the centre of another rotating circle (epicycle), and you get epicycloids just in the case where they have exactly the same spatial speed (r₁ω₁ = r₂ω₂) while rotating in the same orientation, which does not in general have to hold.
An epitrochoid: to see correspondence to an epicycle, imagine the black centre of the rolling circle tracing out another larger circle, which is the deferent in the Ptolemaic model; the distance between the black centre and the red tracing point is the radius of the epicycle.
That's why I wanted at least part of this general class of curves to be discovered early. As epitrochoids and hypotrochoids are different only in whether the generating circle is rolled inside or outside a larger one, this is an effort of making the philosophical comprises relative to Greek commitments about natural motion ‘not worse’ than the Ptolemaic system. As epicycles, hypotrochoids involve rotation in the opposite sense instead, and ellipses are actually a particularly simple kind of hypotrochoid, so in some senses it is perhaps philosophically better (but there is a velocity complication). The Tusi couple is an example of how one might start exploring them from a relatively simple beginning.

The pattern is more general; a particularly pretty case is caused by the 13:8 mean motion resonance of Venus.
Also, a similar model to the elliptic case has been developed by Ismaël Boulliau (1645), in which the elliptical orbit is directly a conic section of certain cone rotating along its axis that passes through the opposite focus than the one occupied by the Sun.
Interestingly, Boulliau was the originator of the inverse-square force law, though he did not believe it. His objection to Kepler's investigation of inverse-distance influence of the Sun was theoretical on the analogy that any such physical influence should, just like intensity of a light ray, diminish with the square of the distance.
 
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Vorpal

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Sidereal Year

According to Aratos (3c BCE), Aristarkhos of Samos had calculated the sidereal year to be 365'4'10'4', which is an encoding of 365+1/(4-1/(10-1/4)) = 365¼+1/152 = 365.25658 days, the result of adding one day every two Kallippic cycles. This is both accurate and convenient, since Aristarkhos was a proponent of heliocentrism. Meanwhile, according to Vettius Valens (2c CE), the Babylonians considered the sidereal year to be 365¼+1/144 = 365.25694 days, which is not attested in any extant Babylonian source. However, from Babylonian System B, one can straightforwardly derive 365¼+1/154 = 365.25648, which is both extremely accurate and closer to Aristarkhos' sidereal year. The value 1/144 might be the result of rounding, since it is a much simpler fraction in the Babylonian sexagesimal system, though probably also a different method.

In Babylonian astronomy, the angular positions are sidereal, along the ecliptic in the direction from Aries to Taurus, rather than relative to the vernal equinox. The common unit of time of the mean synodic month, or more frequently its 1/30th, the mean lunar day. For the Sun's and Moon's daily displacement along the zodiac, the Babylonian tables mostly used an apparently round figure, but the most precisely stated ones I could find were in one unusual extant tablet (BCM A.1845-1982.2, also No.97 in Ossendrijver's Procedure Texts), which attests a very precise n = 0;59,08,09,48,40°/d for the Sun, producing a sidereal year of 360°/n = 6,5;15,33,45,39 d = 365.259378 d. For the Moon, it gives n = 13;10,34,51°/d for a sidereal month of 27;19,18,04. ... This is both less accurate and much farther from both Aristarkhos and the Babylonian figure according to Vettius Valens; therefore, I will discount these values and simply use the usual System B values instead.

According to System B, the immediately relevant mean lunar months are:
Synodic: 29;31,50,8,20 d = 765433/25920 d​
Sidereal: 27;19,18 d = 16393/600 d​
This value of the synodic month is also equal to the molad of the Hebrew lunisolar calendar, since it ultimately derives from the Babylonians, and was was probably first derived by Aristarchos as an intermediately rounded 1778037/60210 in order to give easier numbers in sexagesimal calculations. It has sub-second accuracy to modern value; the moon's orbital period and the Earth's rotation both slightly slow, so the ratio of the mean synodic month to the mean solar day is a bit more stable than may be expected, though it also distorts. The denominator is connected to the Babylonian še (barleycorn) and the Hebrew halakim (parts/portions): the Babylonians divided the day in 360 (degrees), and each uš into 72 še, while the Hebrews the day into 24 hours and each hour into 15×72 = 1080 halakim, so 1 še = 1d/(25920) = 1 helek = 3⅓ s.

The commonly used values in System B give substantially more accurate results than than the precise values of the previous unusual tablet, i.e. it is a better estimate of the sidereal month, and indeed later Islamic astronomers (al-Khwārizmī, Ibn al-Kammād, Ibn al-Raqqām) had a sidereal month of 27;19,18,00,10 d. Since the Earth goes around the Sun once per sidereal year, or the other way around, there should be one more sidereal month than synodic month in a sidereal year, so 1/(sid.yr) = 1/(sid.mn) - 1/(syn.mn), and the System B values give 365¼d+9m19s, or to the nearest še,
Sidereal year: 365¼ d + 168 še (9m20s) = 9467448/25920 d = 365.25648 d,​
which matches a projected mean sidereal year for the Seleucid era when most of those observations were taken, differing from the modern value of 365.25636 (365¼d+9m10s) primarily due to a different duration of the mean solar day as Earth's rotation slows.
The result 6,5;15,23,17,28,..., which can be rounded to the nearest še in order to fit to the same denominator as the synodic lunar month as 6,5;15,23,20. On the other hand, if it is instead rounded as 6,5;15,25, the result would be exactly what Vettius Valens attests the Babylonian sidereal year to be.

Based on analysis of historical records of eclipses, the mean solar day increases on average by 1.78 ms/century, which is slower than the 2.3 ms/cy predicted by tidal effects, with some additional variation around that average. The ephemeris day of 86400 SI seconds is ultimately based on the mean solar day ca. 1820 and most Babylonian observations come from the Seleucid period, and we can expect that mean solar day was shorter by roughly 35 ms ca. -300 and by 30 ms ca. -100, from the envelope graph in [Stephenson et al. (2003)], from spline fits in [Stephenson et al. (2016)].

The mean sidereal year in atomic time can be calculated as the inverse of the rate of change of the mean longitude λ, i.e. λ˙ = λ₁ + 2λ₂t + 3λ₃t² + O(t³), where the coefficients are in arc-seconds and t is in Julian millennia from J2000 [Simon et al., §5.8.3]:
λ₁ = 1295977422.83429, λ₂ = −2.04411, λ₃ = −0.00523.​
Putting them together, we can estimate the smoothed sidereal year in terms of the mean solar day by year:
-300: 365¼d + 9m22.3s​
-200: 365¼d + 9m21.3s​
-100: 365¼d + 9m20.5s​
-000: 365¼d + 9m20.0s​
In the years 100-600, the resulting estimate has seconds fluctuate between roughly 19.3-19.7.
Here 25920/168 ≈ 154 compares to Aristarchos' 152.

The moral of the story is that Babylonian astronomy once again could achieve an extreme level of accuracy (though they did so with Greek figures).
Edit: Corrected using spline fit from newer [Stephenson et al. (2016)] paper.
 
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Rufus Shinra

Well-known member
The moral of the story is that Babylonian astronomy once again could achieve an extreme level of accuracy.
Well, wasn’t there a pretty big motivation to get it as right as possible, if only to be able to predict the events that would impress one’s ruler?
 

Vorpal

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Well, wasn’t there a pretty big motivation to get it as right as possible, if only to be able to predict the events that would impress one’s ruler?
Yes, but with caveats. It is so especially for Babylonian astronomy, which was primarily concerned about predicting the timings of characteristic phenomena: heliacal risings and settings, oppositions, etc. This is in some contrast to Greek astronomy, which is more modern in the sense that the problem the Greeks were trying to solve is predicting the geocentric angular positions of celestial bodies given an arbitrary time. On the other hand, the Babylonians had clever interpolation and other techniques to predict positions at a particular time, including one equivalent to the trapezoidal rule of numerical integration to get positions of planets from velocities [ref]. (The Oxford development referred to is the Merton rule of the Oxford calculators, i.e. mean speed theorem that Galileo later used in analysis of uniformly accelerated motion, and hence is directly one of the shoulders that Newton stood on.)

According to Stephenson's reconstruction of the smoothed mean solar day, the System B prediction may be correct to a fraction of a second for the late Seleucid period. I simply didn't expect this level of accuracy, so I got very excited. :) Stephenson's method should be good, e.g. if a total eclipse is recorded in Babylon but modern modeling of constant day says it should be in north-west Africa instead, that's a good indication of cumulative deviation from atomic time, and so forth.

To see how unusually accurate this heliocenric method based on System B is, compare its 365.25648 d to some year lengths of other ancient astronomers, which I mostly copy from Gomez' Aristarchos of Samos [ref]:
Code:
Aratos' List (3c BCE):
Euktemon, Philip   365¼+1/76    = 365.26316
Aristarchos        365¼+1/152   = 365.25658 ⋆
A Babylonian       365¼-1/284   = 365.24648
Sudines            365¼-1/304   = 365.24671
?                  365¼+3/4868  = 365.25062
?                  365¼+1/1596  = 365.25063
Vettius Valens' List (2c CE):
Meton, Euktemon    365¼+1/76    = 365.26316
Aristarchos        365¼-15/4868 = 365.24692
Chaldeans          365¼-1/284   = 365.24648
Babylonians        365¼+1/144   = 365.25694 ⋆
Other:
Censorinus         365¼+1/1623  = 365.25062
Hipparkhos         365¼-1/300   = 365.24667 (via Ptolemy)
Hipparkhos         365¼+1/288   = 365.25347 (via Galen)
Hipparkhos?        365¼+1/144   = 365.25694 ⋆ (guess)
The figures below 365¼ could be interpreted as the tropical year and above as the sidereal year; Aristarchos was probably the first astronomer to realise they are different, followed by Hipparkhos. Sexagesimally simple fractions like 1/144 and 1/288 suggest approximation, so are more likely to have been different in original form. ... Galen's Hipparkhos figure is probably wrong; Hipparkhos is known to have claimed that the precession of the equinoxes is at least 1°/cy, and this value was widely adopted since Ptolemy, and Galen's number is inconsistent with it. Meanwhile, 365¼+1/144 would work since 360°(1-(365¼-1/300)/(365¼+1/144)) = 1.013°/cy. It also means that it's possible that Vettius Valens misattributed the same figure.

I was pretty surprised by it for several reasons. Partly because the general level of acceptable accuracy in thoses times was generally to a quarter-day. One could say that's about how much you had to get right before the ruler considers you a substandard astronomer, and a lot of the observation tables don't bother being more precise than that. Partly because it is much more accurate than any known attested value of any astronomer anywhere near those times; perhaps because the astronomers weren't expected to do better than ¼d, they could round things more. Finally, also because I got it stuck in my head that the mean solar day deviates by 2.3 ms/cy, which is an oft-repeated prediction of tidal friction, but is actually wrong and over-compensates the correction.

Ultimately, it actually makes sense that the mean lunar periods to be highly accurate despite all that, as the astronomers had access to very long-reaching astronomical records that could average out errors: on the extreme end, astronomers even through the medieval times generally used regnal year of Nabonassar for dates, from 747 BCE, simply because that was about how far back they had records. And since this particular method of calculating the sidereal year is predicted from lunar observations, which are generally of much higher quality than solar ones, it makes sense for it to be accurate too. I guess it shouldn't be that surprising that Aristarchos, one of the few heliocentrists, would come closest to it either.

Reputedly, the chief source of error in Hipparkhos' figures are a misunderstanding of different conventions of when the day starts in different astronomical tables, noon or midnight. It really makes me wonder what he would have gotten if these dating corrections were done properly. ... I am intrigued by this because the correct long-term axial precession value is (ed.) 1°0′21″/(72 years) ≈ (360°/25920)/yr, very fitting with the Babylonian and Hebrew denominator scheme. So it might be possible that even if the tables don't give a very good value for the tropical year, that if one ‘rounds like a Babylonian’ so to speak, one may coincidentally hit upon an accurate value.
 
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