Editing Titius–Bode law (section)

==Blagg formulation==
In 1913, [[Mary Adela Blagg|M. A. Blagg]], an Oxford astronomer, revisited the law.<ref name=blagg>
{{cite journal
 |last=Blagg |first=M. A. |author-link=Mary Adela Blagg
 |year=1913
 |title=On a suggested substitute for Bode's law
 |journal=[[Monthly Notices of the Royal Astronomical Society]]
 |volume=73 |pages=414–422
 |doi=10.1093/mnras/73.6.414 |doi-access=free
|bibcode=1913MNRAS..73..414B }}
</ref>
She analyzed the orbits of the planetary system and those of the satellite systems of the outer gas giants, Jupiter, Saturn and Uranus. She examined the log of the distances, trying to find the best 'average' difference.

[[File:Function f of Blagg formulation of Titius Bode Law.png|thumb|upright=1.6| {{center|The empirical correction function &thinsp;{{mvar|f}}&thinsp; introduced in Blagg's reformulation of the Titius–Bode law.}}]]

Her analysis resulted in a different formula:

<math display="block">\ \mathsf{distance} = A \cdot \bigl[\ B + f\left(\alpha + n\ \beta \right)\ \bigr]\ \bigl(\ 1.7275\ \bigr)^n ~.</math>

Note in particular that in Blagg's formula, the law for the Solar System was best represented by a progression in {{math|1.7275}}, rather than the original value {{math|2}} used by Titius, Bode, and others.

Blagg examined the [[Moons of Jupiter|satellite system of Jupiter]], [[Moons of Saturn|Saturn]], and [[Moons of Uranus|Uranus]], and discovered the same progression ratio {{math|1.7275}}, in each.

However, the final form of the correction function {{mvar|f}} was not given in Blagg's 1913 paper, with Blagg noting that the empirical figures given were only for illustration. The empirical form was provided in the form of a graph (the reason that points on the curve are such a close match for empirical data, for objects discovered prior to 1913, is that they ''are'' the empirical data).

Finding a formula that closely fit the empircal curve turned out to be difficult. [[Fourier analysis]] of the shape resulted in the following seven term approximation:<ref name=blagg/>

<math display="block">\begin{align}
\ f \bigl(\ \theta\ \bigr) \;=\; 0.4594 \;+\;\; & 0.396\ \cos\!\bigl(\ \theta - 27.4^{\circ}\ \bigr) \;+\; 0.168\ \cos\!\bigl(\ 2\ (\ \theta - 60.4^{\circ})\ \bigr) \;+\; 0.062\ \cos\!\bigl(\ 3\ (\ \theta - 28.1^{\circ})\ \bigr) \;+\;\\
\;+\;\; & 0.053\ \cos\!\bigl(\ 4\ (\ \theta - 77.2^{\circ})\ \bigr) \;+\; 0.009\ \cos\!\bigl(\ 5\ (\ \theta - 22^{\circ})\ \bigr) \;+\; 0.012\ \cos\!\bigl(\ 7\ (\ \theta - 40.4^{\circ})\ \bigr) ~.
\end{align}</math>

After further analysis, Blagg gave the following simpler formula; however the price for the simpler form is that it produces a less accurate fit to the empirical data. Blagg gave it in an un-normalized form in her paper, which leaves the relative sizes of {{mvar|A}}, {{mvar|B}}, and {{mvar|f}}&thinsp; ambiguous; it is shown here in normalized form (i.e. this version of &thinsp;{{mvar|f}}&thinsp; is scaled to produce values ranging from {{math|0}} to {{math|1}}, inclusive):<ref name=review>
{{cite journal
 |last1=Lobban |first1=G. G.
 |last2=Roy    |first2=A. E.   |author2-link=Archie Roy
 |last3=Brown  |first3=J. C.
 |date=October 1982
 |title=A review of Blagg's formula in the light of recently discovered planetary moons and rings
 |journal=[[Journal of the British Astronomical Association]]
 |volume=92  |number=6  |pages=260–263
 |bibcode=1982JBAA...92..260L
 |url=https://ui.adsabs.harvard.edu/link_gateway/1982JBAA...92..260L/ADS_PDF
}}
</ref>

<math display="block">\ f \bigl(\ \theta\ \bigr) \;=\; 0.249 \;+\; 0.860 \ \left( \frac{\ \cos\ \Psi\ }{\ 3 - \cos\!\left(\ 2\ \Psi\ \right)\ } \;+\; \frac{ 1 }{\ 6 - 4\ \cos\!\left(\ 2\ \Psi - 60^{\circ} \right)\ } \right)\ ,</math>

where <math>\ \Psi \equiv \theta - 27.5^{\circ} ~.</math>

Neither of these formulas for function {{mvar|f}} are used in the calculations below: The calculations here are based on a graph of function {{mvar|f}} which was drawn based on observed data.

{| class="wikitable" style="text-align:right;"
|+ Constants for Blagg's refinement of the Titius–Bode law<br/>(as modified by {{harvnb|Nieto|1970}})
|-
! System   !! {{mvar|A}} !! {{mvar|B}} !! {{mvar|&alpha;}} !! {{mvar|&beta;}}  
|-
|style="text-align: left"| [[Solar System|Sun-orbiting bodies]] &thinsp; || &emsp; 0.4162 &emsp; || &emsp; 2.025{{0}} &emsp; || &emsp; 112.4° &emsp; || &emsp; {{0}}56.6° &emsp;
|-
|style="text-align: left"| [[Moons of Jupiter]]     || &emsp;  0.4523         || &emsp;  1.852{{0}} || &emsp;  113.0° || &emsp;  {{0}}36.0°
|-
|style="text-align: left"| [[Moons of Saturn]]      || &emsp;  3.074{{0}}     || &emsp;  0.0071     || &emsp;  118.0° || &emsp;  {{0}}10.0°
|-
|style="text-align: left"| [[Moons of Uranus]]      || &emsp;  2.98{{0}}{{0}} || &emsp;  0.0805     || &emsp;  125.7° || &emsp;  {{0}}12.5°
|}

Her paper was published in 1913, and was forgotten until 1953, when [[Archie Roy|A. E. Roy]] came across it while researching another problem.<ref>
{{cite journal
 |last=Malcolm |first=Roy 
 |year=1955
 |title=Is Bode's law a coincidence?
 |journal=[[Astounding Science Fiction]]
 |volume=LV |issue=5
}}
</ref>
Roy noted that Blagg herself had suggested that her formula could give approximate mean distances of other bodies still undiscovered in 1913. Since then, six bodies in three systems examined by Blagg had been discovered: [[Pluto (dwarf planet)|Pluto]], [[Sinope (satellite)|Sinope]] ([[Moons of Jupiter|Jupiter&nbsp;IX]]), [[Lysithea (satellite)|Lysithea]] ([[Moons of Jupiter|J&nbsp;X]]), [[Carme (satellite)|Carme]] ([[Moons of Jupiter|J&nbsp;XI]]), [[Ananke (satellite)|Ananke]] ([[Moons of Jupiter|J&nbsp;XII]]), and [[Miranda (moon)|Miranda]] ([[Moons of Uranus|Uranus&nbsp;V]]).

Roy found that all six fitted very closely. This might have been an exaggeration: out of these six bodies, four were sharing positions with objects that were already known in 1913; concerning the two others, there was a ~6% overestimate for Pluto; and later, a 6% underestimate for Miranda became apparent.<ref name=review/>

===Comparison of the Blagg formulation with observation===

Bodies in parentheses were not known in 1913, when Blagg wrote her paper. Some of the calculated distances in the Saturn and Uranus systems are not very accurate. This is because the low values of constant {{mvar|B}} in the table above make them very sensitive to the exact form of the {{nobr| function &thinsp;{{mvar|f}}&thinsp;. }}

{|
|- style="vertical-align:top;"    <!-- table to contain / align tables -->

| <!-- first col. table container table -->

{| class="wikitable"  style="text-align:right;vertical-align:center;margin-left:0.6em;"
|+ [[Solar System|Sun-orbiting bodies]]
|-
! [[Solar System|Planet]] / [[minor planet|minor&nbsp;pl.]] !! {{mvar|n}} !! obs. dist. !! Blagg pred.
|-
|style="text-align: left"| [[Mercury (planet)|Mercury]] || −2 || 0.387 || 0.387
|-
|style="text-align: left"| [[Venus (planet)|Venus]] || −1 || 0.723 || 0.723
|-
|style="text-align: left"| [[Earth (planet)|Earth]] || 0 || 1.000 || 1.000
|-
|align=left| [[Mars (planet)|Mars]] || 1 || 1.524 || 1.524
|-
|style="text-align: left"| [[Vesta (minor planet)|Vesta]] ||rowspan="4"| 2 || 2.362 ||rowspan="4"| 2.67{{0}}
|-
|style="text-align: left"| [[Juno (minor planet)|Juno]] <!-- || 2 --> || 2.670 <!-- || 2.67{{0}} -->
|-
|style="text-align: left"| [[Pallas (minor planet)|Pallas]] <!-- || 2 --> || 2.774 <!-- || 2.67{{0}} -->
|-
|style="text-align: left"| [[Ceres (dwarf planet)|Ceres]] <!-- || 2 --> || 2.769 <!-- || 2.67{{0}} -->
|-
|style="text-align: left"| [[Jupiter (planet)|Jupiter]] || 3 || 5.204 || 5.200
|-
|style="text-align: left"| [[Saturn (planet)|Saturn]] || 4 || 9.583 || 9.550
|-
|style="text-align: left"| [[Uranus (planet)|Uranus]] || 5 || 19.22{{0}} || 19.23{{0}}
|-
|style="text-align: left"| [[Neptune (planet)|Neptune]] || 6 || 30.07{{0}} || 30.13{{0}}
|-
|style="text-align: left"| [[Pluto (dwarf planet)|(Pluto)]] || 7 || (39.48) || 41.8{{0}}{{0}}
|}

| <!-- next col. table container table -->

{| class="wikitable"  style="text-align:right;vertical-align:center;margin-left:0.3em;"
|+ [[Moons of Jupiter]]
|-
! [[Moons of Jupiter|Jupiter satellite]] !! {{mvar|n}} !! obs. dist. !! Blagg pred.
|-
|style="text-align: left"| [[Amalthea (moon)|Amalthea]]|| −2 || 0.429 || 0.429
|-
| style="text-align: center"|'''{{color|silver|—}}''' || −1 || style="text-align:center"|'''{{color|silver|—}}''' || 0.708
|-
|style="text-align: left"| [[Io (moon)|Io]]|| 0 || 1.000 || 1.000
|-
|style="text-align: left"| [[Europa (moon)|Europa]] || 1 || 1.592 || 1.592
|-
|style="text-align: left"| [[Ganymede (moon)|Ganymede]]|| 2 || 2.539 || 2.541
|-
|style="text-align: left"| [[Callisto (moon)|Callisto]]|| 3 || 4.467 || 4.467
|-1
| style="text-align:center"|'''{{color|silver|—}}''' || 4 || style="text-align:center"|'''{{color|silver|—}}''' || 9.26{{0}}
|-
| style="text-align:center"|'''{{color|silver|—}}'''  || 5 || style="text-align:center"|'''{{color|silver|—}}''' || 15.4{{0}}{{0}}
|-
|style="text-align: left"| [[Himalia (moon)|Himalia]] ||rowspan="3"| 6 || 27.25{{0}} ||rowspan="3"| 27.54{{0}}
|-
|style="text-align: left"| [[Elara (moon)|Elara]] <!-- || 6 --> || 27.85{{0}}  <!-- || 27.54{{0}} -->
|-
|style="text-align: left"| ([[Lysithea (moon)|Lysithea]]) <!-- || 6 --> || (27.85)  <!-- || 27.54{{0}} -->
|-
|style="text-align: left"| ([[Ananke (moon)|Ananke]]) ||rowspan="4"| 7 || (49.8){{0}} ||rowspan="4"| 55.46{{0}}
|-
|style="text-align: left"| ([[Carme (moon)|Carme]]) <!-- || 7 --> || (53.3){{0}} <!-- || 55.46{{0}} -->
|-
|style="text-align: left"| [[Pasiphae (moon)|Pasiphae]] <!-- || 7 --> || 55.7{{0}}{{0}} <!-- || 55.46{{0}} -->
|-
|style="text-align: left"| ([[Sinope (moon)|Sinope]]) <!-- || 7 --> || (56.2){{0}} <!-- || 55.46{{0}} -->
|}

| <!-- next col. table container table -->

{| class="wikitable" style="text-align:right;vertical-align:center;margin-left:0.3em;"
|+ [[Moons of Saturn]]
|-
! [[Moons of Saturn|Saturn satellite]] !! {{mvar|n}} !! obs. dist. !! Blagg pred.
|-
|style="text-align: left"| ([[Janus (moon)|Janus]]) || −3 || (0.538) || 0.54{{0}}
|-
|style="text-align: left"| [[Mimas (moon)|Mimas]] || −2 || 0.630 || 0.629
|-
|style="text-align: left"| [[Enceladus]] || −1 || 0.808 || 0.807
|-
|style="text-align: left"| [[Tethys (moon)|Tethys]] || 0 || 1.000 || 1.000
|-
|style="text-align: left" [[Dione (moon)|Dione]] || 1 || 1.281 || 1.279
|-
|style="text-align: left"| [[Rhea (moon)|Rhea]] || 2 || 1.789 || 1.786
|-
| style="text-align:center"|'''{{color|silver|—}}''' || 3 || style="text-align:center"'''{{color|silver|—}}''' || 2.97{{0}}
|-
|style="text-align: left"| [[Titan (moon)|Titan]] || 4 || 4.149 || 4.140
|-
|style="text-align: left"| [[Hyperion (moon)|Hyperion]] || 5 || 5.034 || 5.023
|-
| style="text-align:center"|'''{{color|silver|—}}''' || 6 || style="text-align:center"|'''{{color|silver|—}}''' || 6.3{{0}}{{0}}
|-
| style="text-align:center"|'''{{color|silver|—}}''' || 7 || style="text-align:center"|'''{{color|silver|—}}''' || 6.65{{0}}
|-
| style="text-align:center"|'''{{color|silver|—}}''' || 8 || style="text-align:center"|'''{{color|silver|—}}''' || 7.00{{0}}
|-
|style="text-align: left"| [[Iapetus (moon)|Iapetus]] || 9 || 12.09{{0}} || 12.11{{0}}
|-
|style="text-align: left"| [[Phoebe (moon)|Phoebe]] || 10 || 43.92{{0}} || 43.85{{0}}
|}

| <!-- next col. table container table -->

{| class="wikitable"  style="text-align:right;vertical-align:center;margin-left:0.3em;"
|+ [[Moons of Uranus]]
|-
! [[Moons of Uranus|Uranus satellite]] !! {{mvar|n}} !! obs. dist. !! Blagg pred.
|-
|style="text-align: left"| ([[Miranda (moon)|Miranda]]) || −2 || (0.678) || 0.64{{0}}
|-
| style="text-align:center"|'''{{color|silver|—}}''' || −1 || style="text-align:center"|'''{{color|silver|—}}''' || 0.77{{0}}
|-
|style="text-align: left"| [[Ariel (moon)|Ariel]] || 0 || 1.000 || 1.000
|-
|style="text-align: left"| [[Umbriel]] || 1 || 1.394 || 1.393
|-
|style="text-align: left"| [[Titania (moon)|Titania]] || 2 || 2.293 || 2.286
|-
|style="text-align: left"| [[Oberon (moon)|Oberon]] || 3 || 3.058 || 3.055
|}

|} <!-- end table container table -->