As I mentioned in this blogpost, in special relativity any observer will measure the speed of light in a vacuum to be , irrespective of whether the observer is moving towards or away from the source of light. We can think of the speed of light as a cosmic speed limit, nothing can travel faster than it.

But, let us suppose that we have two reference frames and moving relative to each other with a speed of , 90% of the speed of light. Surely, if someone in frame fires a high-speed bullet at a speed of , an observer in frame will think that the bullet is moving away from him at a speed of , which seemingly violates the comic speed limit.

What have we done wrong?

We cannot simply add velocities, as we would do in Newtonian mechanics. In special relativity we have to use the Lorentz transformations to add velocities. How do we do this? Let us remind ourselves that the Lorentz transformations can be written as

The Lorentz transformations to go either from reference frame , or to go from .

Calculating a velocity in two different reference frames

To calculate the velocity of some object moving with a velocity in reference frame we need to use these Lorentz transformations.

We start off by writing

and

We will now take the derivative of each term, so we have

and

We can now write (the velocity of the object as seen in frame ) as

The terms cancel, and dividing each term on the right hand side by gives

where was the velocity of the object in reference frame .

Going back to our example of and , we can see that the velocity as measured by an observer in reference frame will be

, not as we naively calculated.

The constancy of the speed of light

What happens if a person in reference frame shines a light in the same direction as is moving away from ? In this case, . Putting this into our equation for we get

So they both agree that the light is moving away from them with the same speed !

As those of you following my blog will know, I am currently on a cruise around New Zealand, giving astronomy talks. One of my six talks is about our current understanding of whether there is (or was) life on Mars. I try to only talk about objects which are visible during the cruise, and Mars is currently visible in the evening sky, albeit a lot fainter than it was in May when it was at opposition.

One of the talks I am giving on this cruise is our current understanding of whether there is (or was) life on Mars.

The question of whether there is life on Mars, or whether there ever has been in its history, is a fascinating one. I thought I would do a series of blogs to explore the question. But, I have to begin by saying that ANY search for life beyond Earth is predicated by our understanding of life on Earth. The only thing, it would seem, required by all forms of life which we have found on earth is water. Extremophiles show that life can exist without oxygen, without light, at high pressure, in radioactive environments; in fact in all sorts of environments which humans would find impossible. But, none of the life so far found on Earth can exist without water.

As a consequence, all searches for life in our Solar System tend to begin with the search for water. Now, it may be that life beyond Earth could have evolved to exist without the need for water. I am no chemist, but I don’t think there is anything particularly unique about water in its chemistry which makes it impossible for living cells to use some other substance. Water is the only substance on Earth which can exist in all three forms naturally (solid, liquid and gas), so it does occupy an unique place in the environment found on Earth. But, on Titan for example, methane seems to exist in all three forms. Maybe life has evolved on Titan to metabolise using methane in the same way that life on Earth has evolved to metabolise using water. We don’t know.

So, I thought I would start this series of blogs with the big news in the 1890s, that Martians had built canals on the red planet!

Schiaparelli and Martian ‘canali’

The Schiaparelli space probe which ESA sadly failed to land on Mars recently was named after Italian astronomer Giovanni Schiaparelli. In the late 1880s he reported seeing ‘canali’ on the surface of Mars. Although this means ‘channels’, it got mis-translated to ‘canals’, and led to a flurry of excitement that this was evidence of an intelligent civilisation on Mars.

The idea grew that Martians had built canals to transport water from the “wet” regions near the poles to the arid equatorial regions. The ice caps of Mars are easily visible through a small telescope, so astronomers had known for decades that Mars had ice caps which they assumed were similar to the ice caps on Earth.

Giovanni Schiaparelli’s map of ‘canali’ on Mars, from 1888.

One person who became particularly taken with this idea of canals on Mars was American Percival Lowell. Lowell came from a rich Bostonian family, and had enough personal wealth to build an observatory in Flagstaff, Arizona. He set about proving the existence of life on Mars, writing several books on the subject. He published Mars (1895), Mars and Its Canals (1906), and Mars As the Abode of Life (1908). But, by 1909 the 60-inch telescope at Mount Wilson Observatory had shown that the ‘canali’ were natural features, and Lowell was forced to abandon his ideas that intelligent life existed on Mars.

However, his Flagstaff Observatory was to go on and make important contributions to astronomy. In the 1910s Vesto Slipher was the first person to show that nearly all spiral nebulae (spiral galaxies as we now call them) showed a redshift, the first bit of observational evidence that the Universe is expanding. And, in 1930 Clyde Tombaugh discovered Pluto at Flagstaff Observatory.

In part 2 of this blog, next week, I will talk about the first space probes sent to Mars, and the first images taken of Mars by a space probe which successfully orbited the planet, Mariner 4.

Last week, my attention was drawn to an article entitled “Here’s Why There Ought to be a Cap on Women Studying Science and Maths”, which actually appeared on the alt-right website Breibart back in June 2015. Breibart is a right-wing site founded by Andrew Breibart, and the website’s current executive chairman is Stephen Bannon, who has recently been named by president-elect Donald Trump as his chief strategist in Trump’s new White House team.

Women: Know Your Limits! (a comedy sketch by Harry Enfield)

“Here’s Why There Ought to be a Cap on Women Studying Science and Maths” is what I can only describe as a diatribe, written by Milo Yiannopoulos, and Englishman who seems to have become one of the main voices for the kind of misogynistic nonsense which many in the “contrastive right” like to spout. I don’t want to give any more attention to the utter drivel that Yiannopoulos writes in his article, but I thought I would quote just a few lines.

Even women who graduate with good degrees in science subjects often don’t use them: they switch careers in their twenties, abandoning the hard sciences. In some cases, they simply drop out of the workforce altogether. This is a disaster for the men who missed out on places, and it’s a criminal waste of public funds.

That’s why I think there ought to be a cap on the number of women enrolling in the sciences, maths, philosophy, engineering… and perhaps medicine and the law, too. It’s hugely expensive to train a doctor, but women have something like a third of the career of a man in medicine, despite having equal access to Harvard Med. Women make up the majority of medical students.

In addition to twisting statistics and misquoting feminist academics to support his theory that women are not really capable of doing science, Yiannopoulos even suggests that there should be caps on women studying law! Interestingly, Yiannopoulos himself has started and failed to complete degrees at two different universities, which I guess means that he is a bit of an expert on failure and wasting taxpayers’ money.

Anyway, there is really no way to argue logically with an imbecile like Yiannopoulos, his views are best countered by a bit of humour. So, here is a wonderful sketch by Harry Enfield – “Women: Know Your Limits!”. Enjoy!

Tomorrow (Friday 25 November) I am boarding a plane which will eventually get me to Brisbane (Australia), via Seoul. Yes, I’m aware that Brisbane is not New Zealand, but in Brisbane I am joining a cruise which is going around New Zealand. The cruise will last for 14 nights, and I will give about 6 talks during the two weeks.

The Princess Cruise leaves Brisbane on 27 November and returns on 11 December. I will be giving astronomy talks on the 14-night cruise.

This will be the 5th cruise which I’ve done with Princess, and the 6th in total. The last time I did a cruise in the southern hemisphere was in February, when I cruised from Buenos Aires to Santiago around Cape Horn. Unfortunately, during that 14-night cruise, we had only one clear night! I am hoping for better weather this time, as in addition to my talks I run star parties to show the guests what is visible in the night-time sky.

Many of the guests will probably be from Europe or the United States, and so will be very keen to see the Southern Cross. I will also show them the Magellanic Clouds if weather permits. The New Moon is on the 29 November, so the first week of the cruise will be ideal to see the Magellanic Clouds if the skies are clear. After that, the brightening moon will render them all by invisible. So, fingers crossed we get some clear skies during the first week!

The other week I was asked to explain how a cylinder (or ball) rolling down a slope differs from e.g. a ball being dropped vertically. It is an interesting question, because it illustrates some things which are not immediately obvious. We all know that, if you drop two balls, say a tennis ball and a cannon ball, they will hit the ground at the same time. This is despite their having very different masses (weights). Galileo supposedly showed this idea by dropping objects of different weights from the tower of Pisa (although he probably never did this, see our book Ten Physicists Who Transformed Our Understanding of Reality).

With a tennis ball and a cannon ball, they clearly have very different masses (weights), but will fall to the ground at the same rate. This fact, contrary to the teachings of Aristotle, was one of the key breakthroughs which Galileo made in our understanding of motion. But, what about if we roll the two balls down a slope? If we build a track to keep them going straight, will a tennis ball roll down a slope at the same rate as a cannon ball? The answer is no, and I will explain why.

Rolling rather than dropping

When a ball rolls down a slope, it starts off at the top of the slope with gravitational potential energy. When it starts rolling down the slope, this gravitational potential energy gets converted to kinetic energy. This is the same as when the ball drops vertically. But, in the case of the ball dropping vertically, the kinetic energy is all in the form of linear kinetic energy, given by

where is the mass of the ball and is its velocity (which is increasing all the time as it falls and speeds up). The gravitational potential energy is converted to linear kinetic energy as the ball drops; by the time the ball hits the bottom of its fall all of the PE has been converted to KE.

If, instead, we roll a ball down a slope, the kinetic energy is in two forms, linear kinetic energy but also rotational kinetic energy, which is given by

where is the ball’s moment of inertia, and is the ball’s angular velocity, usually measured in radians per second. The key point is that the the moment of inertia for the two balls in this example (a tennis ball and a cannon ball) have a different value, because the distribution of the mass in the two balls is different. For the tennis ball it is all concentrated in the layer of the rubber near the ball’s surface, with a hollow interior. For the cannon ball, the mass is distributed throughout the ball.

Two cylinders rolling down a slope

Let us, instead, consider the case of two cylinders rolling down a slope. One is a solid cylinder, the other is a hollow one with all of its mass concentrated near the surface. We will make the two cylinders have the same mass; this can be done by making the material from which the hollow cylinder is made denser than the material for the solid cylinder. So, even though the material of the hollow cylinder is all concentrated near the surface of the cylinder, and there is a lot less of it, if it is denser it can have equal mass.

A solid cylinder on an inclined plane. We will make the mass of this solid cylinder the same as that of the hollow cylinder, by making it of less dense material. Although it will have the same mass and the same radius , it will not have the same moment of inertia .

We will start both cylinders from rest near the top of the slope, and let them roll down. We will observe what happens.

A hollow rolling down an inclined plane. We will make the hollow cylinder denser than the solid one, so that they both have the same mass and the same outer radius . But, they will not have the same moment of inertia .

When things are dropped, the rate at which they fall is independent of the mass, but when they roll the rate at which they roll is not indpendent of the moment of inertia. In particular, it is not independent of the distribution of mass in the rolling object. As this video shows, the solid cylinder rolls down the slope faster than the hollow one!

But, why??

Why does the solid cylinder roll down quicker?

The reason that the solid cylinder rolls down faster than the hollow cylinder has to do with the way that the potential energy (PE) is converted to kinetic energy. Because the cylinder is rolling, some of the PE is converted to rotational kinetic energy (RKE), not just to linear kinetic energy (LKE). The only way that a cylinder can roll down a slope is if there is friction between the cylinder and the slope, if the slope were perfectly smooth the cylinder would slide and not roll.

The torque (rotational force) is related to the angular acceleration in a similar way that the linear force is related to linear accelerate . From Newton’s second law we know that where is the mass of the object. The rotational equivalent of this law is

where is the moment of inertia. The moment of inertia is different for a hollow cylinder and a solid cylinder. For the solid cylinder it is given by

where is the mass of the cylinder and is the radius of the cylinder. For the hollow cylinder, the moment of inertia is given by

where are the outer and inner radii of the annulus of the cylinder. We are going to make the hollow cylinder such that the inner 80% is hollow, so that . We will make the mass of the two cylinders the same.

Thus, for the hollow cylinder, we can now write

The cylinder accelerates down the slope due to the component of its weight which acts down the slope. This component is where is the acceleration due to gravity and is the angle of the slope from the horizontal. To make the maths easier, we are going to set , as .

Friction always acts in the opposite direction to the direction of motion, and in this case the friction is related to the torque via the equation

so we can write

where is the rotational acceleration. Re-arranging this to give , we have

The force down the slope, is just the component of the weight down the slope minus the frictional force acting up the slope.

The angular acceleration is given by where is the linear acceleration. So, we can re-write Eq. (3) as

Now we will put in the moments of inertia for the solid cylinder and the hollow cylinder. For the solid cylinder, we can write

The mass can be cancelled out, and assuming , we have

Notice that Equation (5) does not have the mass in it, as this cancels out. It also does not have the radius of the cylinder in it; the acceleration of the cylinder as it rolls down the slope is independent of both the mass and the radius of the cylinder.

For the hollow cylinder, again using Eq. (4), we have

This simplifies to

As with Equation (5), Equation (6) is independent of both mass and radius.

So, as we can see, the linear acceleration for the hollow cylinder is 2.69 m/s/s, less than the linear acceleration for the solid cylinder, which was 3.27 m/s/s. This is why the solid cylinder rolls down the slope quicker than the hollow cylinder! And, the result is independent of both the mass and the radius of either cylinder. Therefore, a less massive solid cylinder will roll down a slope faster than a more massive hollow one, which may seem contradictory.

Summary

All objects falling vertically fall at the same rate, but this is not true for objects which roll down a slope. We have shown above that a solid cylinder will roll down a slope quicker than a hollow one. This is because their moments of inertia are different, it requires a greater force to get the hollow cylinder turning than it does the solid cylinder. Remember, the meaning of the word ‘inertia’ is a reluctance to change velocity, so in this case a reluctance to start rolling from being stationary. A larger moment of inertia means a greater reluctance to start rolling.

The solid cylinder will start turning more quickly from being stationary than the hollow cylinder, and this means that it will roll down the slope quicker. This result is independent of the masses (and radii) of the two cylinders; even a less massive solid cylinder will roll down a slope quicker than a more massive hollow one, which may be counter-intuitive.

New images of the European Space Agency’s Beagle 2 have emerged recently, suggesting that it came closer to success than has long been thought. These new images have been analysed more thoroughly and carefully than previous images of Beagle 2, and with the help of a computer simulation it is being suggested that Beagle 2 did not crash land. Instead, this team led by Professor Mark Sims of Leicester University are arguing that Beagle 2 deployed, but not completely correctly. They suggest that, due to not deploying correctly, that it may well have done science for a period of about 100 days, before shutting down due to lack of power. They even suggest that there is a very slim possibility that it is still working.

I do have to take issue, however, with the way this story is worded on the BBC website. It implies that we now know, with certainty, that Beagle 2 operated for some period on the surface of Mars. This is not true. One study has argued that it did. One swallow does not make a summer. This particular team’s analysis and study will need to be looked at by others before we can say with any reasonable certainty that Beagle 2 survived its landing.

New images of Beagle 2 taken by NASA’s Mars Reconnaissance Orbiter have been analysed by a computer model, suggesting it may have actually worked for a short period of time.

As with any suggestion which flies in the face of conventional wisdom, this claim will need to be checked and followed up by others. But, if the evidence is sufficiently strong that Beagle 2 did not crash, then it will come as a relief to those who worked on it and have long felt that it failed in a crash. Sadly, even if it did work, we have not received any data back from it; and that is not going to change.

Following on from my blog “Is Tim Peake getting younger or older?” , a bit of fun to work out whether time was passing more slowly or more quickly for Tim Peake in the ISS than it is for us on the ground, it got me thinking about the global positioning system (GPS) that so many of us use on a daily basis. Whether it is using a SATNAV in our car, or a GPS-enabled watch to measure how far and fast we have run, or using maps on a smartphone, GPS must be one of the most-used satellite developments of the last few decades.

As I blogged about here, communication satellites need to be at a particular height above the Earth’s surface so that they orbit the Earth in the same time that it takes the Earth to rotate. In addition to their altitude, communication satellites can only orbit the Earth about the equator, no other orientation will allow the satellite to hover in the same place relative to a location on Earth.

But what about the satellites used in GPS? What kind of an orbit are they in?

The GPS satellites’ orbits

It turns out that the GPS satellites are not in a geo-stationary orbit, but are in fact in an orbit which leads to their orbiting the Earth exactly twice in each sidereal day (for a definition of sidereal day see my blog here).

The GPS system consists of 31 satellites in orbit around the Earth

We can work out what radius from the Earth’s centre this needs to be by remembering that the speed of orbit is given by

where is the speed of orbit, is the universal gravitational constant, is the mass of the Earth and is the radius of orbit from the centre of the Earth (not from its surface).

A sideral day is 23 hours and 56 minutes, which in seconds is seconds. So, half a sidereal day is seconds. We will call this the period . The speed of orbit, is related to the period via the equation

where is the radius of the orbit, the same as in equation (1), and is just the circumference of a circle. So, squaring Equation (1), we can write

So, in terms of we can write

In terms of height above the Earth’s surface, we need to subtract off the radius of the Earth, so the altitude, which I will call , is going to be

Why are GPS satellites in this kind of an orbit?

As I didn’t know what kind of an orbit GPS satellites were in before I wrote this blog, the next obvious question is – why are they in an orbit which is exactly half a sidereal day? It is clearly not coincidental! To answer this question, we need to first of all discuss how GPS works.

GPS locates your position by measuring the time a signal takes to get to your GPS device from at least four satellites. Your device can identify from which satellites it gets a signal, and the system knows precisely the position of these satellites. By measuring the time the signals take to you reach you from each of the satellites, it is able to calculate how far each one is from you, and then by using triangulation it can work our your location. There are currently 31 satellites in the system, so often there are more than four visible to your GPS device. The current 31 satellites have all been launched since 1997, the original suite of 38 satellites launched between 1978 and 1997 are no longer in operation.

As I mentioned in my blog about geostationary satellites, a satellite in a geostationary orbit can only orbit above the Earth’s equator. This would clearly be no good for a GPS system, as all the satellites would lie to the south of someone in e.g. Europe or North America. As I said above, there are currently 31 operational satellites; the 31 are divided into 6 orbital planes. If there were 30 satellites this would be 5 in each orbit. The orbits are inclined at to the Earth’s equator. Each orbit is separated from the other one by 4 hours (equivalent to ) in longitude.

As one can see approximately 6 hours in right ascension to both the east and west of one’s location, this means that there will be at least 3 of the orbits above the horizon, and sometimes more. If there were 5 satellites in each orbit this would mean that each one would pass a particular latitude 4 hours before the next one. So, at any particular time there should be some satellites further north than one’s location and some further south, as well as some further east and some further west. This configuration allows for the necessary triangulation to obtain one’s location.

The orbits are inclined at to the equator and separated by 4 hours (equivalent to ) in right ascension, as this diagram attempts to show

Is the time-dilation effect due to SR or GR more important for these satellites?

We already showed in this blog that, for the International Space Station, the time-dilation due to Special Relativity (SR) has a greater effect on the passage of time than the time-dilation due to General Relativity (GR). What about for the GPS satellites?
The speed of orbit for the GPS satellites at a radius of from the Earth’s centre is, using Equation (1),

As we showed in my blog about Tim Peake, the speed of someone on the Earth’s surface relative to the centre of the Earth is , so the relative speed between a GPS satellite and someone on the Earth’s surface is given by

Compare this to the value for the ISS, which was , it is less than half the speed.

This value of leads to a time dilation factor in SR of

which means that the time dilation due to SR is negligible.
The time dilation due to GR is given by (see my blog here on how to calculate this)
, or 22 parts in . Compare this to the ISS, where it was about 1 part in . Clearly the GR effect for GPS satellites is greater, by about a factor of 5, than it was for the ISS. But, conversely, the SR time-dilation effect has become negligible.

To conclude, the time dilation for GPS satellites is nearly entirely due to General Relativity, and not due to Special Relativity. Time is passing more quickly for the clocks on the GPS satellites than it is for us on Earth, the converse of what we found for the ISS, which is in a much lower orbit.

Because the timings required for GPS to work are so precise, the time dilation effect due to GR needs to be taken into account, and is one of the best pieces of evidence we have that time dilation in GR actually does happen.