by Tom Smith
10/28/02
from
QuarkBurger Website
In midOctober of 2002, I stumbled
across a chat room about Surviving Planet X. This got my
interest, so I came in and asked "What's Planet X?". The host pasted
a summary and a bunch of links explaining that Planet X
orbits every 3600 years, and will swing past the Earth within 14
million miles, causing global devastation.
Immediately I was skeptical, as were others in the room. But I
wanted to look at this in an objective, and scientific way.
I was close to completing a Masters in Physics, and had recently
taken three semesters of orbital mechanics, so I decided to run the
numbers.
All objects in space (and on Earth for that matter) follow physical
laws. And it turns out the laws of orbital mechanics are actually
simpler than many other areas in physics because drag is usually
negligible, and planetary bodies, asteroids, stars, etc, do not have
the ability to alter their course like a space craft would. By using
the laws of gravity to determine motion, the movement of objects in
space can be determined far into the future with a high degree of
accuracy.
Let's look at Kepler's Laws. They are the foundation of
orbital mechanics and are found in the first couple of pages in any
orbital mechanics book.

First Law
The orbit of each planet is an
ellipse, with the sun at a focus.

Second Law
The line joining the planet to
the sun sweeps out equal areas in equal times.

Third Law
The square of the period of a
planet is proportional to the cube of its mean distance from
the sun.
Since 1609, these laws have held true
and have survived basically unchanged for centuries.
(There are two exceptions though:
The first law is now "The orbit of
each planet is a conic section (ellipse, parabola or hyperbola),
with the sun at a focus".
The additional orbit types are for
objects going too fast for the sun to pull it back again. These
objects will continue beyond the solar system. The second exception
is in extreme cases where velocities approach the speed of light or
super massive objects such as
black holes are involved. Neither
of these exceptions apply here.)
Using these laws, and the equations they translate into, we can
actually determine quite a bit using four pieces of information
gained from the ZetaTalk website:
1. The period, or time it takes
for a complete orbit, is 3657 years.
2. Encounter date of MayJune of 2003
3. Planet X's orbit is inclined at 32 degrees from
Earth's orbit.
4. Planet X is a
brown dwarf
For clarity, here is an orbit diagram
which shows the basic properties of an elliptical orbit, and the
relationship of Earth's and Planet X's orbit.
Note the elongated shape of the
ellipse. This is determined by the eccentricity of the orbit. An
eccentricity of 0 is a circle, and the closer it gets to 1.0, the
more stretched out it will be. An eccentricity of 1.0 would no
longer be an ellipse though, that's a parabola. The eccentricity and
the semimajor axis are both important in determining the size and
shape of the orbit.

First we shall determine the
semimajor axis, and this comes directly from Kepler's 3rd law.
T is the period (3657
years). From this we can solve for a, the
semimajor axis which comes to 237 astronomical units (AU). One
AU is the distance from the sun to the Earth.

Second we'll find the
eccentricity. From the diagram above, p is the
semilatus rectum. It's a strange name, but basically it is the
distance to the point on the ellipse above and below the focus
(where the sun is). This is shown on the diagram above as
pt1, and pt2. The significance of these
points is that with Planet X's orbit inclined, these points are
the only points where an encounter with Planet X and
Earth can occur.
The left equation computes p,
but we already know what p is. Since p is where the
encounter occurs, it must be the distance from the sun to the
Earth, or 1 AU. So we solve for e (eccentricity), which comes to
0.99789. This would be a highly eccentric orbit. Then the
perihelion distance and aphelion distance (closest and furthest
point in the orbit) can be determined to be 0.5AU and 474AU
respectively.
How far is Planet X
now?
We can also determine how far
away Planet X is based on the number of days until the
encounter date. For the purposes of this report, let's use May 31st,
2003 as the encounter date, and for the current time, let's use
October 31, 2002. That comes to 212 days. Based on orbital mechanics
that are a bit more tricky, (click image below
right) one can calculate how far an object will travel in
it's orbit over a given period of time. Using this method, I have
determined that on 10/31/02, Planet X would have to be 3.35
AU from Earth
if it is going to be here by May/June of 2003.
There is no getting around this result. For example if a ball is
going to fall from a ledge and hit the ground in 1 second (excluding
air drag effects) the ledge must be 32 feet high, since things on
Earth fall at 32 feet/second squared. It's just the way it works.
So
based on a distance of 3.35AU, and that Planet X is reported to
be a brown dwarf (which are typically about 160,000km in
diameter), we can also compute the angular diameter of Planet X
in our sky, which would be arctan(160,000km/500,000,000km), or 66
seconds of arc. In comparison, Venus is approximately 60 seconds
of arc. Planet X should then appear to be slightly larger than
Venus, and would be the brightest object in the sky after the sun
and moon.
The fact that we can not see Planet X should create serious doubt
that it will encounter Earth in MayJune of 2003. I predict
that the encounter date will come and go without incident, and
ZetaTalk will then issue a new date based on some new piece of
information they just received. The bottom line is, before you
accept something like this as fact, and especially before you
go spreading the story, use your head, and check into it.
Click image right for the complete
method used to derive these results.
P.S. Update
On 1/1/03 Planet X should be 2.57 AU away, and 85 seconds of arc.
This would appear 43% larger than Venus.
