Hey Brady, I am Matt Henderson. I like
making mathematical animations and I
wanted to show a few that relate to the
patterns of orbits in the solar system.
The first one is the inner planets; and
so this is the approximate pattern that
the inner planets trace out - obviously
things aren't to scale. We've got the Sun,
we've got Mercury, Venus, Earth and Mars
going around in their approximate orbits
and relative speeds. So they're spinning
around, the Sun is at the centre and now
we're to actually move so we're
following Earth. And we're interested in,
what are the patterns that are traced out by
the other planets? From kind of like from
the perspective of Earth we start to see
some interesting patterns.
The pattern from Venus is starting to be
quite regular.
We'll start to see this kind of
five-pointed shape;
you can see we've got one of the points
here, we've got two, three, and the next
time Venus comes around it completes
this five points: 1, 2, 3, 4, 5.
And that's because the ratio of the
orbits between Venus and Earth are- is in
an 8 to 13 ratio so the difference there
is accounting for the five-fold symmetry.
In general it's just like a pretty pattern.
I think these are called epicycloids,
they're kind of like the spirograph
patterns that you get rolling circles
around circles.
And those just happen to be the ones
that the planets trace out.
Now let's look at this asteroid Cruithne,
that has been referred to as Earth's
second moon, but that's just a
sensationalist kind of thing. It is-
what it is is an asteroid that's in
orbital resonance with the Earth
which means- in this case it means that
its year is the same as Earth's year;
but it's not orbiting Earth, it's also
orbiting the Sun but in a more kind of
elliptical orbit. So if we mistakenly
thought it was orbiting orbiting the
Earth then what would its orbit look
like? It wouldn't be a circle, it'd be
some kind of
mix of circles and ellipses or something;
so let's see.
So again we start out with the Sun at
the centre;
Earth is orbiting and then this is Cruithne,
this little asteroid that is orbiting
with the same year.
Now we've shifted to the perspective of
the Earth and the path that Cruithne is
tracing out is this kind of kidney bean
shape - again this these aren't the exact
correct orbits but it's roughly
kind of like what's going on. So we don't
see an elliptical orbit; if we were wrong
and we thought Earth was the centre of
the solar system then Cruithne would
be the sort of weird second moon tiny
asteroid guy that is orbiting us in this
kind of kidney bean shape. - (Brady: That would)
(take some explaining if you believe that)
(the Earth was at the centre of)
(everything!) - Yeah that's a kind of
complicated behaviour to explain,
but I suppose it's just built up of a
circle and an ellipse kind of added
together, so maybe as soon as you saw
that you might invent that explanation.
(You put these on Twitter but what have)
(you done in the past?) - Yeah I used to run
this blog on Tumblr that- where I posted
mathematical GIFs and
back then I don't think they were
quite as good as my new ones, I think I've
sort of progressed a little bit. So
for example, you know, this was the new
Cruithne animation, this is the old one
here. It's explaining the same
shape but it takes a little bit longer
to see exactly what's happening; possibly
because you don't have that shift in
perspective, you know where you establish
the orbits with the Sun at the centre
and then you shift to just following
Earth because Earth is stationary.
You can kind of see what's going on, right?
(The Sun's stationary too in that one)
Yeah the Sun is stationary as well. So if
we look at the new one, the Sun is also
still stationary right, but the fact that
the Sun is at the centre of the old one
is maybe a bit confusing as well. Here
I've just put Earth is the centre to
emphasise that we're doing a sort of
geocentric thing. In both I've done the
trick of having like stars in the
background, which are just randomly
generated points, but the stars that sort
of swing around I think helps to show
you're in a sort of weird reference
frame that's rotating with the Earth.
(And then you've got another old one too from there?)
Yeah, related to this ratio of
Venus and the Earth: got this one here
which is, you have Venus going around,
you've got Earth going around, and you
just draw a line between the two. It's a
bit different to tracing the relative
path but you get a similar shape, you get
that five-pointed shape that I pointed
out before. - (It's fun stuff all this)
(planets isn't it?) - Yeah, and I mean for me
it's kind of just a way to talk about
the spirograph patterns, epicycloids,
that you get from circles rotating
around. It's just kind of fun to plug
in the numbers that roughly correspond
to like Venus and Earth for example. And,
of course, it's not exactly 8 to 13 ratio
but it gives the 5 pointed symmetry
because 8 plus 5 is 13.
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