Monday, 25 January 2016

Journey to the Far-Side of the Sun

There was a movie, in the old days, Journey to the Far-Side of the Sun (also known as Doppleganger) which (spoiler alert) posits that there is a mirror version of the Earth hidden on the other side of the Sun, sharing the orbit with our Earth. The idea is that this planet would always be hidden behind the Sun, and so we would not know it there there.

This idea comes up a lot, over and over again. In fact, it came up again last week on twitter. But there's a problem. It assumes the Earth is on a circular orbit.

I won't go into the details here, but one of the greatest insights in astronomy was the discovery of Kepler's laws of planetary motion, telling us that planets move on elliptical orbits. With this, there was the realisation that planets can't move at uniform speeds, but travel quickly when closer to the Sun, while slowing down as their orbits carry them to larger distance.
 There has been a lot of work examining orbits in the Solar System, and you can simply locate the position of a planet along its orbit. So it is similarly simply to consider two planets sharing the same orbit, but starting at different locations, one at the closest approach to the Sun, one at the farthest.

Let's start with a simple circular orbit with two planets. Everything here is scaled to the Earth's orbit, and the circles in the figures coming up are not to scale. By here's an instance in the orbit.

It should be obvious that at all points in the orbit, the planets remain exactly on opposite sides of the Sun, and so would not be visible to each other.

So, here's a way of conveying this. The x-axis is the point in the orbit (in Earth Years) while the y-axis is the distance a light ray between the two planets passes from the centre of the Sun (blue line). The red line is the radius of the Sun (in Astronomical Units).
The blue line, as expected, is at zero. The planets remain hidden from each other.

Let's take a more eccentric orbit, with an eccentricity of 0.1. Here is the orbit
This doesn't look too different to the circular case above. The red circle in there is the location of the closest approach of each line of sight to the centre of the Sun, which is no longer a point. Let's take a look at the separation plot as before. Again, the red is the radius of the Sun.
Wow! For small segments of the planets orbits, they are hidden from one another, but for most of the orbit, the light between the planets pass at large distances from the Sun. Now, it might be tricky to see each other directly due to the glare of the Sun, but opportunities such as eclipses would mean the planets should be visible to one another.

But an eccentricity of 0.1 is much more than that of the Earth, whose orbit is much closer to a circle with an eccentricity of 0.0167086 . Here's the orbit plot again.
So, the separation of the paths between the planets pass closer to the centre of the Sun, but, of course, smaller than the more eccentric orbits. What about the separation plot?
Excellent! As we saw before, for a large part of the orbits, the light paths between the planets passes outside the Sun! If the Earth did have a twin in the same orbit, it would be visible (modulo the glare of the Sun) for most of the year! We have never seen our Doppleganger planet!

Now, you might complain that maybe the other Earth is on the same elliptical orbit but flipped so we are both at closest approach at the same time, always being exactly on the other side of the Sun from one another. Maybe, but orbital mechanics are a little more complex than that, especially with a planet like Jupiter in the Solar System. It's tugs would be different on the Earth and its (evil?) twin, and so the orbits would subtly differ over time.

It is pretty hard to hide a planet in the inner Solar System!

Sunday, 3 January 2016

Throwing a ball in a rotating spaceship

A long time ago, I wrote a post about the Physics of Rendezvous with Rama, a science fiction story by Arthur C. Clarke set on an immense alien spaceship. The spaceship rotates, providing the occupants with artificial gravity, a staple of science fiction. I mentioned in the article that I am not an immense fan of a lot of science fiction, as much of it relies on simple "magic", but Clarke knew his physics and so he knew that the "gravity" experienced in the rotating ship will differ to that on Earth, and previously I wrote about what happens if you jump off a cliff.

In the last week, there was a question on twitter (it's an internet thing) about the movie Elysium which has a spectacular rotating spacecraft with the Earth's rich abroad.
While it's a shame that the plot was not as spectacular, the question was how can such a station keep it's atmosphere.

This is an interesting question, as you might think that it would all simply zip off into space. But the key point is that the atmosphere itself is also rotating with the ship, and so "experiences gravity". But what happens to individual molecules in the air?

Well, we can do a bit of physics (yay!) and work this out. If you have done some basic physics (or chemistry) you have have encountered the ideal gas laws, relating some of the key properties of a gas, such as the temperature, volume and pressure. The derivation of these can be deceptively simple, but the equations are very powerful. And to do these derivations, you essentially assume that atoms are very bouncy balls that bounce off the walls of the vessel.

So, what about bouncing a ball inside a rotating space station? There is a nice little discussion here on the key physics, but it is very simple (if you are an undergrad who as done your classical mechanics, you should give this a go). The important point is that seen from an external observer, a bouncing ball follows a straight-line path (remember, there are no forces acting on the ball), but the view to a person inside the ship will be different. Here's a simple example.

So, this is a bouncing ball as seen by an outside observer (taking into account the motion of the ship in the collisions).
The blue is the wall of the ship (which is rotating) and the red is the path of a bouncing ball.

What about the view from inside?
Boing, boing, boing! The ball bounces off into the distance, and, if we leave it, will bounce all the way around the station and hit you in the back of the head :)

So, an air molecule will bounce in a similar fashion around the station, and so an atmosphere will be kept there too. Yay for all the rich people!!

But, being good physicists, we can start to play with the velocity and direction of our molecules (I'm playing with the velocity relative to the outside observer. It's easy to consider it relative to the velocity of the ship at the stating point).

Slowing the ball down gives the same path to the outside observer, but why does the internal observer see?
Oh... The ball bounces the other way around the ship. Cool!

What if we lob the ball into the air rather than bouncing it off the all of the shop. In fact, let's arrange for it to go straight up (again, velocity relative to the outside observer). We'll adjust the velocity so that bounce off the other side and get bak to the start position in the time it takes for a single revolution of the ship.

Again, the red is the path as seen by the outside observer, the black by the inside.
The ball arcs behind the thrower, over the top, and comes back in front of the thrower. In fact, there are several black loops present, each on top of the other.

What if we slow the ball down a little, so the observer on the ship makes two revolutions in the time it takes the ball to get over and back.
Wow!! So we see the ball bounce of the other side of the ship and do some mid-air pirouettes. Let's slow it down by a factor of two again!
And again!
Excellent. Imagine watch this ball fly through the air!

Let's instead double the speed. What do we get?
Again, think about being a being at rest watching the ball bouncing through the ship!

Doubling again!
We can see where this is going!

And if we modify the velocity across the ship so there is not a "resonance" between the bouncing and the rotation, we get
Again, the ball will perform exquisite arcs through the ship and with bounces of the wall!

How much fun is this! I've got to take a break right now, but later I will consider what happens if you want to play a ball game, like tennis, inside a rotating space ship. The rich people might get more than they are asking for!