More Maths

Maths is an essential tool in the Astronomer’s Tool kit. A lot of people are frightened of maths but, viewed simplistically, maths is like French or German. It’s another language. To understand the language of maths you just have to learn a few basics and you can still get by. Obviously, the more you practice your language skills then the better you’ll become. You’ll improve further if you spend time with natives of those lands.

I’ve provided several of the most useful equations below. I’ve also included some explanatory text to help you understand the relevance of these equations.

Isaac Newton's Gravity

This is our first equation. Sir Isaac Newton developed this equation to calculate the force between two masses. The same equation can be applied to calculate the force between the planets, or the stars, or between the Earth and an apple.

Let’s take a look at the equation and see what use we can put it to.

First, let’s calculate the force on you due to the planet Earth.

Let’s say you weight around 80 kg at the Earth's surface. We can look up the mass of the Earth (5.9742 × 10²⁴ kg), the radius (d = 6357km) and the value of the Gravitational constant, G (6.674 x 10^-11 )

Stick the numbers into a calculator and you get an answer of 789 Newtons, Newtons being the units of Force. Yes, they’re named after Sir Isaac.

Now let’s repeat the calculation for the planet Jupiter. Jupiter’s mass is 1.898 x 10²⁷ kg, and it’s 740.52 million kilometres from us. This gives us a force of 0.00000018505847 Newtons.

That’s very small indeed. Very small.

So how can a planet so far away influence our actions here on Earth? Go ask your friendly local astrologer and see if they have a good answer.

We could go one step further and see if we can calculate the force on us from something more mundane, say, my iMac, which is sitting next to me on the desk. It weights around 10 kg and is about half a meter away. What’s the force on me? Put the values into the calculator and the result is 0.000000213568 Newtons. Compare this force to that exerted by the planet Jupiter. It’s twice the number.

An astrologer might counter this simple argument with some hand-waving mumbo-jumbo involving magic crystals and ancient forces but, here’s a proof.

Your PC pulls on you more than Jupiter.

A Simple Pendulum

Here's another simple equation, the equation for a simple pendulum. This one you can actually prove for yourself. You’ll need a piece of string and a weight, and a stop watch.

Take the pendulum and tie it to a door frame. Measure the length of the string in meters and then pull the pendulum to one side. At the precise moment you let go of the string, press start on the stopwatch. Now count how long it takes to swing back and forth ten times. On the tenth count, stop the stopwatch. Repeat this a couple of times and take the average reading. It’s not precise but this is only a demonstration.

We’re going to calculate the rate of gravitational acceleration acting on you at this moment. Don’t know what this is? If we were to take a one kilogram object to the top of a tall building and drop it then (neglecting air resistance) it would accelerate at g meters per second per second.

Firstly, let’s re-arrange the equation to make g the subject.

Now, I used a length of string 1.2m in length. I measured ten swings in 22 seconds, which is 2.2 seconds for one swing.

Plug this into the above and we get:

Now look this value up on the internet. You should see that the internationally agreed value varies from place to place on the planet but the average is around 9.81 ms^-2

Newton's Second Law of Motion

We’re now going to use another equation, Newton’s second law of motion, which says that a force upon an object causes it to accelerate according to the formula F = ma.

Suppose we have a satellite that weighs 500 kg sitting atop a rocket that also weighs 500kg. How hard do we have to push the rocket to overcome gravity and push the satellite into space? Easy. We just multiply the mass of the rocket and the satellite by the acceleration from gravity, which we calculated in the step above:

F = (500 + 500) x 9.78

= 97800 Newtons

That's how much force we have to apply just to defeat the force of gravity holding the satellite on the ground. We have to apply even greater forces if we're to lift the satellite into orbit.

Putting a satellite into Orbit

"It's not rocket science!"

Well, it is.

Suppose we want to launch our 1000 kg satellite into space. We light the blue touch paper and stand well back. The chemicals in the rocket - either solid or liquid - burn, producing an exhaust gas, which is ejected from the back of the rocket, pushing the rocket forwards.

However, what happens if we don't push hard enough? The rocket (and our expensive satellite) fall back to Earth. How hard do we have to push? How fast does the rocket have to be moving before we can put it into orbit?

Rocket engineers give this speed a special name - the escape velocity of the rocket. By modifying the equation for the Simple Pendulum discussed above. we can derive an equation for calculating the escape velocity. Turns out this number depends wholly upon the mass and radius of the planet and not at all upon the mass of the satellite and the rocket combined. How strange is that?

We can look up the value for the mass and radius of the planet Earth and use them to calculate the Escape Velocity for our satellite.

So we can see that a satellite has to be moving at 11 kilometres per second (roughly 7 miles per second) just to go into orbit, and faster still if it's to go somewhere else in the solar system.

You could repeat the same calculation for the Moon. The mass of the Moon is 7.34 x 10^22 kg and its radius is 1700 km. The Gravitational Constant remains the same so plugging the values into the above equation gives an Escape Velocity for the Moon of just 2.38 km/s, which is much lower.

Perhaps you can see why we're so keen to use the Moon as a platform for launching bigger, more powerful rockets. It takes significantly less fuel to push a rocket to its escape velocity on the Moon!

Bode's Law and the Planets

Bodes’ Law is a very useful mathematical tool. It was derived by two mathematician’s Johann Bode and Johann Titius although most astronomers call it Bode’s Law.

What is Bode’s Law? Well, Johann Bode noticed that if you graph the number of a planet against the estimated distance from that planet to the Sun then you get a straight line.

This works reasonably for Mercury, Venus, Earth, Mars, Jupiter and Saturn. The deviation from ideal is not huge, of the order of a couple of percentage points.

Now, at the time Bode and Titius came up with their equation, the planet Uranus was not known. It had been seen on many occasions but few astronomers recognised it for what it was - a genuine planet.

William Herschel was the first to do that. Bode’s Law, although empirically designed, strongly suggested that there might be another planet out beyond Saturn. A search began and, as mentioned, Herschel found it although he did think he was looking at a comet rather than a planet.

With Uranus discovered, astronomers again turned to Bode’s law, which predicted another planet beyond Uranus. Astronomers around the world began the search for this elusive planet although they were unsuccessful.

The French Astronomer Urban Le Verrier noted small but significant deviations in the orbit of Uranus and, working with British Astronomers John Couch-Adams and Astronomer Royal George Biddell-Airy, began searching in what he thought was the right area of the sky. Alas, fate intervened. The story goes that Couch-Adams and his assistant were preparing for an evening of observing when Mrs. Couch-Adams insisted that they delay their search and, instead, join the family for dinner. Later on, after they’d eaten, Couch-Adams returned to the observatory only to find that the sky had clouded over. They’d missed their opportunity. Worse, the planet Neptune was discovered nine days later by the German astronomer, Johann Gottfried Galle. This caused an enormous stink in the British newspapers, which were very critical of Airy’s leadership, maintaining that this should have been a British discovery and not German.

Maths is an incredibly powerful tool and yet so many people are utterly terrified of the subject. I hope the above examples have served to whet your appetite and maybe dig a little deeper into the subject.