Phil./Hist./Chem. 380 Ancient and Modern Science
    5 Parables from the History of Astronomy 1750-1920
    Henry Mendell

    The five parables

  1. Confirmation of a 'false' theory (Bode's Law, Uranus and the planetoids
  2. The observation of a planet (Uranus and Neptune).
  3. Confirmation of theory by theory and observation and conflict between theory and observation:  using a 'false' theory and a 'good theory' to discover a planet (Newton, Bode, and Neptune).
  4. Conflict between theory and observation:  attempts to resolve conflict (Newton and the Perihelion of Mercury).
  5. How important is consistency (the size of the universe and the total gravitational force on a hat).
Bode's Law (on the distance of a planet from the sun)
Named for Johan Elert Bode (1747-1826)
Proposed by Johann Daniel Titius of Wittenburg in 1772
Take the series
0, 3, 6, 12, 24, 48, 96, 192, *
Then the distance of the n'th planet from the sun is:
4 + the n'th member of the series
or
 4 + 3*(2n-1 if n > 1, 0 otherwise)
Planet
Rank
AU
millions of km
Calculation by Bode's Law
Bodes Law
AU by Bodes Law
Mercury
1
0.39
57.9
4 + 3 * 0
4
0.4
Venus
2
0.7
105.2
4 + 3 * 1
7
0.7
Earth
3
1
149.6
4 + 3 * 2
10
1.0
Mars
4
1.5
227.9
4 + 3 * 4
16
1.6
???
5

4 + 3 * 8
28
2.8
Jupiter
6
5.2
778.3
4 + 3 * 16
52
5.2
Saturn
7
9.5
1427
4 + 3 * 32
100
10.0
Uranus: William Herschel observed Uranus 13 March 1781 and first announced it as a comet. Only later did he observe its circular movement. Named it Georgium Sidus (so-called in England until 1850). Bode proposed Uranus. Uranus (19.2) confirms Bode's Law (19.6 AU).

The Gap at 5: Bode's Law predicts a planet at 2.8 AU. So F.X. von Zach in 1785 actually proposed one. Guiseppe Piazzi discovered Ceres (2.77 AU) on 1 Jan. 1801, while Heinrich Olbers rediscovered it 1 Jan 1802. On 28 March 1802, Olbers discovered Pallas (2.77 AU). Hence Olbers proposed (on the basis of their similar orbits 1681.4 and 1683.3 days) that they are fragments of a larger planet. Juno (2.67 AU) was soon discovered by C.L. Harding and Vesta (2.36 AU by Olbers) soon followed. The planetoids confirm Bode's Law.

Planet
Rank
AU
millions of km
Calculation by Bode's Law
Bodes Law
AU by Bodes Law
Mercury
1
0.39
57.9
4 + 3 * 0
4
0.4
Venus
2
0.7
105.2
4 + 3 * 1
7
0.7
Earth
3
1
149.6
4 + 3 * 2
10
1.0
Mars
4
1.5
227.9
4 + 3 * 4
16
1.6
Planetoids?
5
~2.77
4 + 3 * 8
28
2.8
Jupiter
6
5.2
778.3
4 + 3 * 16
52
5.2
Saturn
7
9.5
1427
4 + 3 * 32
100
10.0
Uranus
8
19.2
2869
4 + 3 * 64
196
19.6

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Moral 1:
To see a planet as a planet one needs to see it moving relative to the fixed stars and to see it as not having a strong elliptical orbit.
Uranus:
  1.  J. Flamsteed saw it as a fixed star in 1690.
  2.  P.C. Lemonnier made eight observations of it during opposition 1768-69.
  3.  There were at least eight other sightings.
  4.  William Herschel  (1781) first announced Uranus as a comet.  But comets, unlike fixed stars encourage one to track the motion, that is, to take a second look.

Moral 2:

Tossing out bad data?  Suppose the theory conflicts with the observation.
 
Neptune:
 Lefrançais de Lalande (1732-1807) observed Neptune, 8 May and 10 May 1795.  Seeing a discrepency, he rejected one as an error and marked the other as questionable.
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  • Parable 3:  Using Theory to Discover Neptune

    • Problem:

    In Alewxis Bouvard's tables for Jupiter, Saturn, and Uranus based on Laplace's Mécanique Céleste, Uranus was off by more than 1' from its observed position.
     

    Possible Solutions

    Reaction
    1.  Reject Newtonian mechanics (NOT A SERIOUS PROPOSAL)
    2.  Reject Laplace's approximate solution to the equations involved in applying Newtonian mechanics to more than 2 bodies.
    		 (This is a mathematical question)
    3.  Reject some observations
    a)  Reject older observations (e.g., Flamsteed's).
    b)  Only accept those observations since Herschel's discovery.
    		 (The error was actually seen to increase)
    4.  Reject the values for the parameters used by Laplace and Bouvard.  E.g., increase the mass of Saturn (F.W. Bessel) (Saturn's mass is now too big for other aspects of "Newtonian" theory)
    5.  Another planet (several people, but first explicitly by Rev. T.J. Hussey in a letter to George Airy (1834), followed by F.B.G. Nicolai (1835), Bessel (1842 in conversation with John Herschel)).  Note that it is in the air.

    The Race and the Methods

    Basic Point

    There is no mathematical solution to the differential equations of the motions of 3 moving bodies in Newtonian mechanics.  All methods for determining the motions of 3 or more bodies involve approximations.
     
     
    1843 Royal Society of Sciences, Goettingen offers 50 ducats for satisfactory solution (papers to be submitted by Sept. 1846)
    1843 John Couch Adams, an undergrad at St. John's, Cambridge, supposes a planet at twice Uranus' distance, fitting Bode's Law (196*2 = 392). Adams' error was actually only 2 degrees off from the planet which then was about 1/2 degree from the ecliptic.  Adams gave his results to George Airy (Sept 1845) and James Challis (Nov. 1, 1845)
    1845 U.J.J. Leverrier gives his results to French Academy, 10 Nov.
    1846 The chase gets hot and a winner will be declared.
    1 June Leverrier publishes a memoir.
    9 July Based on Adam's predictions, Airy suggests to Challis a search 30 deg. (long.) x 10 deg. (lat.).
    4 August Challis (Northumberland) searches through 2 sweeps with comparisons with known charts.
    31 August Leverrier presents a paper, again supposing Bode's Law.
    18 Sept. Leverrier writes to John G. Galle (1812-1910), chief assistant at Berlin, to look for a disk > 3'.
    ~23 Sept. The letter was received and J.F. Encke, the director approves the search.  H.L. d'Arrest, a student living at the observatory suggests after first night that they look at new charts being then prepared by Berlin Academy.  An eight magnitude star, now visible, was not on the chart.  The next night it was observed to have moved.

    Dominique Arago to whom Leverrier had delivered his communication named the planet Neptune, after people outside France objected to the name Leverrier.

    Methods and Ironies:

     

    Leverrier
    Adams I
    Adams II
    Actual
    Semi-major axis
    36.154
    38.38
    37.27
    Eccentricity
    0.1076
    0.16103
    0.112062
    0.008533
    Long. of perihelion
    284°45'
    315°57'
    299°11'
    43°45'
    Mean longitude
    328°47'
    325°8'
    323°2'
    Epoch
    1 Jan. 1847
    1 Oct. 1846
    1 Oct. 1846
    1 Oct. 1846
    True longitude
    326°32'
    328°
    329°
    327°57

     
    1. One needs to assume some value for the distance of the unknown planet.  Both Adams and Leverrier start with 38.8 AU from Bode's Law.  In fact it is closer to 30 AU.  Hence, they derive a much more elliptical orbit than Neptune actually has (it is very circular) and so approximate the segment of motion for 1780-1845, when its influence on Uranus is greatest.
    2. Bode's Law is then in conflict with the assumption that Neptune is a planet and ÅgobservationsÅh.  What is involved in these observations?
    Planet
    Rank
    AU
    millions of km
    Calculation by Bode's Law
    Bodes Law
    AU by Bodes Law
    Mercury
    1
    0.39
    57.9
    4 + 3 * 0
    4
    0.4
    Venus
    2
    0.7
    105.2
    4 + 3 * 1
    7
    0.7
    Earth
    3
    1
    149.6
    4 + 3 * 2
    10
    1.0
    Mars
    4
    1.5
    227.9
    4 + 3 * 4
    16
    1.6
    Planetoids?
    5
    ~2.77
    4 + 3 * 8
    28
    2.8
    Jupiter
    6
    5.2
    778.3
    4 + 3 * 16
    52
    5.2
    Saturn
    7
    9.5
    1427
    4 + 3 * 32
    100
    10.0
    Uranus
    8
    19.2
    2869
    4 + 3 * 64
    196
    19.6
    Neptune 9
    30
    4498
    4 + 3 * 128
    388
    38.8
    Pluto
    10
    39.44
    5900
    4 + 3 * 256
    772
    77.2

    Is there a way one could save Bode's Law?

    A Rosicrucian book once seen but, alas, not carefully noted by the professor of this course claimed that there are only seven planets and that Neptune and Pluto are not planets, because they violate Bodes Law, but  instead are "celestial wanderers".  Note that the word "planet" is from Greek and means "wanderer"!

    This is what Imre Lakatos used to call "monster barring", preserving a law about some class A by claiming that the exceptions to the law are not members of A because they violate the law in question.

    The discovery of Neptune is generally regarded as the refutation of Bode's Law.
     

    More important,

    It is regarded as one of the greatest confirmations of the predictive power and fecundity of Newtonian mechanics.

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    Problem:

    1845:  Leverrier discovers (from transits of the sun) that the perihelion of Mercury progresses 35" more per century than expected from the gravitation of other planets.

    By the end of century this value was more like 43"

    Choices (before Einstein's Theory of General Relativity):

    1. 1. Other unknown masses of matter act on Mercury.
    2. 2. "the intensity of gravitation does not precisely follow Newton's law."


    1a. Intra-Mercurial planets, from an observation of dark transit of sun (Leverrier).  In fact, by 1870 was thought to be an illusion.

  • The eclipses of 1900 and 1905 observed at Harvard and Lick (7-8 mag) failed to reveal any such bodies.
    • 1.b. The intra-Mercurial planets are too small to be seen.  If so they would have to be so numerous as to diffuse light (e.g. a zodiacal light).  They would then, if on the ecliptic, produce a movement of the node. 1.c.
    i.  Intra Mercurial planets have mean inclination 9° and mean node 48° long (Simon Newcome, 1895).  If between Mercury and Venus, inclination 7.5° and 35° node, mass 1/37000000.
    Newcombe proposed that the sun's gravitation is not exactly as the inverse square.  Newcombe showed that Mars advances by 5" per century more than expected and PREDICTS 1.5" for the perigee of the moon.
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    Assumption:  The universe is infinite and matter is everywhere.

    Assumption:  The force of gravity is GMm/R2.

    Problem: What is the net gravitational force on a hat?  Infinite or 0 or what?

    Consider the infinite set of objects opposite to each other and each 1 meter apart affecting the hat in the following configuration.
     


     

    Each weight has a force of attraction with the hat = Gm, and so the net force = 0

    Now consider the same configuration.  Since we suppose the weights are infinite, we can take them like this.

    Taking each weight on the left matched with two on the right:

    Net Gravitational Force (towards the right) = infinite.

     

    How to React:

    1. Reject Newtonian Mechanics?
    2. Reject the Infinitude of the Universe?
    3. Ignore it as a mathematical puzzle whose solution we do not now know (and may never know)
    4. Treat it as a research project.