### Re: Can I solve for the tatol alignment of the 9 planets & sun by using math?

Date: Tue Aug 21 12:30:04 2001
Posted By: Erika Gibb, Grad student, Physics & Astronomy/Origins of Life, RPI
Area of science: Astronomy
ID: 997826615.As
Message:
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Vithia,

How easy it is to calculate the alignments of planets depends strongly on
your mathematical background.  In middle school, I assume you have not gone
much past algebra.  If you know at least some trigonometry (you need an
understanding of angles) then you should be able to do relatively simple
calculations to estimate times for alignments using known periods for the
planet's orbits.  The more planets you try to include, the more challenging
the calculation.  Also, alignments are not perfect in astronomy - to be
considered and alignment it just needs to be close.  For more information,
see the sunspot web site given below.  If you are good with computers, you
may be able to write a program to do it for you.  I'll give you an example
to illustrate the process.  In this example I've rounded the numbers for
simplicity (I've also assumed circular orbits which is a close
approximation).

Example: Mercury goes around the Sun once every 88 days and Earth's orbital
period is about 365 days.  You can not simply say that every 88 days Earth
and Mercury line up because while Mercury went around the Sun, Earth was
also moving.  Earth moves approximately 1 degree around its orbit each
day.  Using this simplification, by the time Mercury gets back to its
starting point (after 88 days), Earth has moved an additional 88 degrees
around its orbit.  Mercury has to catch up before we have another
alignment.  Mercury moves about 360/88=4.1 degrees per day.  Hence, the
time it takes for Mercury to catch up to Earth after completing its 88 day
orbit is given by

omega_Mercury = 4.1 omega_Earth
omega_Mercury * time = omega_Earth * time + 88 degrees

where omega is the angular velocity or number of degrees per day traveled
by a planet, and time is the time it takes for the planets to line back
up.  The 88 comes from the additional number of degrees traveled by Earth
while Mercury went once around its orbit.  Solving for time gives 28 days.
This means it takes the 88 day Mercury period, plus an additional 28 days
for a total of 116 days for Mercury to line back up with Earth.  This is
known as its synodic period and is the value you will most likely want to
use in your calculations.  The synodic periods of the planets are (in days)

115.9 Mercury
583.9 Venus
779.9 Mars
398.9 Jupiter
378.1 Saturn
369.7 Uranus
367.5 Neptune
366.7 Pluto

Now, an alignment of both Mercury and Venus with Earth occurs about once
for every 5 of Mercury's synodic periods (115.9 goes into 583.9 about 5
times) or once every 1.6 years.  After working out a few examples, it
should start to become obvious that the more planets you include, the
longer the time period between alignments.

Now you have an idea of how to calculate the times between alignments.  If
you would like to estimate a date for an actual alignment, you still need a
starting point.  You can either take the positions of planets as they
currently are in their orbits (see the links below) and include those in
your calculations or take the time of the last major planetary alignment
and start from there.  If you use the planet's current positions, you will
need to add a term into your calculations to take their current positions
into account.

Hopefully this gives you enough information to decide if this is a feasible
science fair project for you.  As I said, it depends strongly on how
comfortable you are with math.  Good luck!

Erika

http://www.fourmilab.ch/solar/solar.html
This site has an interactive orrery, which allows you to see the
orientation of the solar system at any time.  You can check your time
estimate for alignments here.  However, since alignments are so rare, you
will not be able to simply guess a time for alignments.  Calculations are
still necessary.
This site shows the orbital positions of the planets at the current time,
looking from above the solar system.  This will give you a starting point
for their positions.
http://www.sunspot.noao.edu/PR/alignment.html
This site has an entire article devoted to discussing planetary alignments,
including how close planets have to be for it to be considered an
alignment.

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