Suppose someone asks you the distance
between the Earth and the Moon, can you answer it? Many of you probably
can answer it in a blink. We know that it takes light slightly more than
one second to travel the Earth-Moon distance and light goes at 300,000
km/s, so the distance between the Earth and the Moon is somewhat larger
than 300,000 km. The actual distance is about 380,000 km. How about Earth-Sun
distance? The answer for this is easy too. It takes light slightly more
than 8 minutes, or about 500 seconds, to travel from the Sun to the Earth.
Then the Earth-Sun distance should be 500 x 300,000 = 150,000,000 km. This
distance is also called one Astronomical Unit or AU. These two questions
are easy because most of us already know by heart the time it takes for
light to travel from the Moon to Earth or Sun to Earth.
How about the distance from the
Sun to one of the planets like Mercury, Mars, Jupiter...? These are much
harder to answer. There are nine planets and it is not easy to remember
the distance of all of them; unless the planets are formed according to
some formula and we just need to remember the formula. Is there a formula
for the planets' distances? Could we use the formula to find other planets?
We shall see shortly.
Since prehistoric time, until late
18th century, six planets including Earth have been known. They are Mercury,
Venus, Earth, Mars, Jupiter, and Saturn. In the last two centuries, three
new planets were discovered; Uranus in 1781, Neptune in 1846, and Pluto
in 1930. Are we all through? Is there no more distant planet to be discovered?
We can't know for sure but at least we can speculate. So, what might a
possible 10th planet be like?
To begin with, where should it be?
Could it be closer to the Sun than Mercury? For thousands of years, the
planet Mercury's orbital was known to precess about the Sun. The Newtonian
answer to Mercury puzzle was that Mercury was slightly affected by a not
yet discovered inner planet which was named Vulcan. Astronomers searched
for the planet Vulcan and did not find it. Then came Einstein. Einstein's
equations of general relativity were applied to the Mercury problem and
the thousand year puzzle was finally solved. Half a century later, in the
late 1960s, the planet Vulcan was found when captain James T. Kirk met
his donkey ear first officer Mr. Spock in the Star Trek series. Unfortunately,
the newly discovered planet does not orbit our Sun, but another sun about
60 light years from us. Just kidding.
The astronomer world today is certain
that the 10th planet would not lie inside the outer most planets. If there
is the 10th planet, it would be farther from the Sun than Pluto is. So
how far should it be from the Sun? For the answer, let's go back to the
18th century.
Back in 1766, a German astronomer,
Johann Daniel Titius, devised a scheme to express the distances of the
planets from the Sun. He did this by starting with a series of numbers,
of which the first was 0, the next 0.3, and each one following double the
one before, thus: 0, 0.3, 0.6, 1.2, 2.4, 4.8, 9.6, 19.2, 38.4, 76.8...
He then added 0.4 to each term in
the series to get the following: 0.4, 0.7, 1.0, 1.6, 2.8, 5.2, 10.0, 19.6,
38.8, 77.2...
The series may be presented with
the formula r(n) = 0.3*2^(n-1)+0.4 with the value of n corresponding to
Mercury = -infinity, Venus =1, Earth = 2 ..., and so on.
The table below shows the Titius's
series comparing with the known distances from the Sun of the six planets
known in Titius's time.
Titius's series Known distance (AU) Planet
.4 .39 1/ Mercury
.7 .72 2/ Venus
1.0 1.00 3/ Earth
1.6 1.52 4/ Mars
2.8
5.2 5.20 5/ Jupiter
10.0 9.54 6/ Saturn
Another German Astronomer named
Johann Bode wrote about Titius's series in 1772 and it has ever since been
referred to as Bode's law. Even with Bode pushing, the series of numbers
was greeted as nothing more than a bit of numerology. But then, in 1781,
a German-born English astronomer named Friedrich Wilhelm Herschel, for
the first time in recorded history, discovered a new planet which was later
called Uranus. And how far is this newly discovered planet from the Sun?
The Bode's law figure for the relative distance of the 7th planet is 19.6
AU and Uranus's actual distance is 19.18 AU. In fact, any astronomer could
have discovered the 7th planet if he had look for it using Bode's law.
Bode's law was suddenly the guide
to fame and new knowledge. To begin with, there was that missing planet
between Mars and Jupiter. At least now they realized there must be a missing
planet, for Bode's law had number 2.8 between the orbits of Mars and Jupiter
and no planet was known to exist there. It had to be searched for.
In 1800, 24 German Astronomers set
up a kind of community effort to find the missing planet. While they were
making all possible preparations, an Italian astronomer, Giuseppe Peazzi,
accidentally discovered the planet in 1801. This planet, named Ceres, is
a very small planet, only 780 km in diameter. It turned out to be only
the first of many hundreds of tiny planets (planetoids) discovered in the
region between Mars and Jupiter in the years since. Planetoids number 2,
3, and 4, by the way, were found by the German team of astronomers within
a year or two after Ceres. Ceres is far the largest of all the planetoids
and its relative mean distance from the Sun is 2.77 AU; Bode's law calls
for 2.8. No astronomer was in the mood to question Bode's law after that.
In 1846, a couple of astronomers,
John Couch Adams of England and Urbain Leverrier of France independently
found the 8th planet, Neptune, using Bode's law in one of their basic assumptions.
The only trouble was that it turned out they had made the wrong assumption.
Neptune should have been at relative distance 38.8 AU from the Sun according
to Bode's law. It wasn't; it was at relative distance 30.1 AU, off by more
than 20%. That blow killed Bode's law and it went back to being nothing
more than an interesting piece of numerology.
When, in 1930, the 9th planet, Pluto,
was discovered at a relative distance 39.5 AU, no one expected it to be
at the Bode's-law distance predicted for the 9th planet. Incidentally,
this fact gave way to speculations whether Pluto could have previously
been a satellite of Neptune until a disruption of the system threw Neptune
out of its previous orbit. What kind of catastrophe that could push Neptune
inward and throw Pluto out of it orbit around Neptune to take its present
wild and eccentric but independently planetary orbit? No one knows.
We can suppose that Neptune, Pluto,
and the satellites form one complex, single planet in which the relationship
of the individual pieces has been confused by catastrophe.
Now let's make up a new and more
complete Titius series.
Titius's series Known distance (AU) Planet
.4 .39 1/ Mercury
.7 .72 2/ Venus
1.0 1.00 3/ Earth
1.6 1.52 4/ Mars
2.8 2.77 4.5/ Ceres
5.2 5.20 5/ Jupiter
10.0 9.54 6/ Saturn
19.6 19.18 7/ Uranus
38.8 30.1+39.5 8,9/ Neptune-Pluto
77.2 ??? 10/ Tenth planet
From the table, the 10th planet
should be 77.2 AU from the Sun. How big would it be? If we consider four
outer gas giant planets (ignoring Pluto), we find a steady decrease in
diameter as we move out from Jupiter. The diameters are 140,000 km (Jupiter),
115,000 km (Saturn), 52,000 km (Uranus), and 44,000 km (Neptune). Carry
that through and let's say the 10th planet has a diameter of about 20,000
km, which is reasonable.
With that diameter and at that distance
from the Sun, the 10th planet should be brighter than the nearer but much
smaller Pluto. Well, then, since Pluto has been discovered and the presumably
larger and brighter 10th planet has not, does that mean the 10th planet
does not exist? Not necessarily. From Kepler's 3rd law, we can calculate
that the period of revolution of the 10th planet would be 680 years, nearly
three times the length of Pluto's period of revolution. It would take a
full year for the 10th planet to shift its position over the width of the
full Moon. This is not the kind of motion that is easily observed by a
casual survey of the heavens. So perhaps it is out there waiting for you
to find it.
Oh boy! I didn't realize the article
would be this long. So it should end soon. But, before we stop, let me
throw in a few final words.
In very recent times, analogous
formulas have been found for the distances of the satellites of Jupiter,
Saturn and Neptune from their mother planets. All this formulas are of
the form r(n) = a*b^n + c They are very similar to Bode's law.
A natural question inevitably suggests
itself at this point. Is it pure coincidence that all these formulas fit
reality or are they rather a reflection of some yet unknown laws of nature?
Bode's law is correct even for objects discovered long after its formulation!
On the one hand, we are aware of the fact that there is always some formula
which fits a few observations, particularly so if the observations are
somehow selected as in the case of the satellites of Jupiter and Saturn,
but on the other, the simplicity of the formulas is striking. In short,
at present astronomy is simply not capable of interpreting these facts.
Titius-Bode's law exists and this cannot be denied, but neither can we
give its explanation now.
Below are the tables of the radii
of the orbits of the satellites of the gas giants. You will easily find
the formulas for the radii of the satellite orbits with a programable hand
calculator. Let me know what you find.
Radii of the orbits of some satellites
of Jupiter (in thousands of km)
Amalthea 181
Thebe 222
Io 422
Europe 671
Ganimedes 1070
Callisto 1880
Radii of the orbits of some satellites
of Saturn (in thousands of km)
Mimas 186
Enceladus 238
Tethys 295
Dione 377
Rhea 527
Radii of the orbits of the satellites
of Uranus (in thousands of km)
Miranda 130
Ariel 191
Umbriel 266
Titania 436
Oberon 583
Duc Ta Vo, Ph.D.
[email protected]
For discussion on this column, join [email protected]
Copyright © 1996 by VACETS and Duc Ta Vo
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