Wonders of the Universe

THE GRAND SCULPTURE

Fish River Canyon in the south of Namibia is one of the world’s great geological features, second only in scale to the Grand Canyon in Arizona, at over 160 kilometres (99 miles) long, 26 kilometres (16 miles) wide and half a kilometre (a third of a mile) deep in places. Like the Grand Canyon, the movement of tectonic plates or volcanic action did not create this scar in Earth’s crust; instead it stands testament to the erosive power of water. The Fish River is the longest river in Namibia, running for over 650 kilometres (403 miles). Despite only flowing in the summer, over millennia it has slowly but forcefully gouged the canyon out of solid rock. This takes energy, and that energy ultimately comes from the Sun as it lifts water from the oceans and deposits it upstream in the highlands to the north. Once the rain begins to fall, gravity takes over. The highlands around the source of the Fish River are at an elevation of over a thousand metres above sea level. When the rain lands on the ground at this elevation, every water droplet stores energy in the form of gravitational potential energy. There is a simple equation that says how much energy each drop has stored up:

U is the amount of energy that will be released if the drop falls from height (h) above sea level down to sea level, m is the mass of the drop and g is the now-familiar acceleration due to gravity – 9.81 m/s².

Every droplet of water raised high by the heat of the Sun has energy, due to its position in Earth’s gravitational field, and this energy can be released by allowing the water to flow downwards to the sea. Some of this energy is available to cut deep into Earth’s surface to form the Fish River Canyon.

The strength of Earth’s gravitational field therefore has a powerful influence on its surface features. This is not only visible in the action of falling, tumbling water, but in the size of its mountains. On Earth, the tallest mountain above sea level is Mount Everest; at almost 9 kilometres (5.5 miles), it towers above the rest of the planet. But Everest is dwarfed by the tallest mountain in the Solar System which, perhaps at first sight surprisingly, sits on the surface of a much smaller planet. Around 78 million kilometres (48 million miles) from Earth, Mars is similar to our planet in many ways. Its surface is scarred by the action of water that once tumbled from the highlands to the seas, dissipating its gravitational potential energy as it fell, although today, the water has left Mars. The planet is only around 10 per cent as massive as Earth, though, so its gravitational pull is significantly weaker, and this is one of the reasons why Mars was unable to hang on to its atmosphere, despite being further away from the Sun. The possibility of liquid water flowing on the Martian surface vanished with its atmosphere, leaving the red planet to an arid and geologically dead future, but Mars’s lower surface gravity has a surprising consequence for its mountains.

Towering over every other mountain in the Solar System is the extinct volcano, Olympus Mons. Rising to an altitude of around 24 kilometres (15 miles), it is almost the height of three Mount Everests stacked on top of each other. The fact that a smaller planet has higher mountains is not a coincidence; it is partly down to environmental factors such as the rate of erosion and the details of the planet’s geological past, but there is also a fundamental limit to the height of mountains on any given planet: the strength of its surface gravity. Mars has a radius approximately half that of Earth’s, and since it is only 10 per cent as massive, a little calculation using Newton’s equation will tell you that the strength of the gravitational pull at its surface is approximately 40 per cent of that on our planet. This changes everything’s weight.

Here on Earth we don’t often think about the difference between mass and weight, but the distinction is very real. The mass of something is an intrinsic property of that thing – it is a measure of how much stuff the thing is made of. This doesn’t change, no matter where in the Universe the thing is placed. In Einstein’s Theory of Special Relativity, the rest mass of an object is an invariant quantity, which means that everyone in the Universe, no matter where they are or how they are moving, would measure the same value for the rest mass.

Weight is different. For one thing, it is not measured in kilogrammes, it is measured in the units of force – newtons. This is easy to understand if you think about how you would measure your weight. When you stand on bathroom scales, they measure the force being exerted on them by you; you can see this by pressing down on them – the harder you push, the greater the weight reading. The force you are exerting on the scales is in turn dependent on the strength of Earth’s gravity. This should be obvious; if I had taken the scales up in the Vomit Comet and tried to stand on them, they wouldn’t have read anything because I would have been floating above them – hence the word ‘weightless’. In symbols, the weight of something on Earth is defined as:

The immense Olympus Mons can exist on Mars because the planet has 40 per cent of Earth’s gravitational pull. However, move this extinct volcano to our planet and it would sink into the ground because of its enormous weight.

W is weight, m is the thing’s mass, and g is the familiar measure of Earth’s gravitational field strength – 9.81 m/s² – with a couple of caveats that we’ll get to below! (For absolute accuracy, the correct definition of weight is; the force that is applied on you by the scales to give you an acceleration equal to the local acceleration due to gravity – i.e. the force the scales exert on you to stop you falling through them.) So, here on Earth a human being with a mass of 80kg weighs 785 newtons; on Mars, the same 80-kg person would weigh approximately 295 newtons.

So your weight depends on a few things; one is your mass, another is the mass of the planet you are on. Your weight would also change if you were accelerating when you measured it, which is another manifestation of the equivalence principle. So, if you took Olympus Mons and stuck it on Earth, then as well as dwarfing every other mountain on the planet, it would also weigh around two and a half times as much as it does on Mars. This enormous force would put its base rock under such intense pressure that it would be unable to support the mountain, so it would sink into the ground. A planet the size of ours cannot sustain a mountain the size of Olympus Mons – it would weigh too much. The highest mountain on Earth, as measured from its base, is Mauna Kea, the vast dormant volcano on Hawaii. It is over one kilometre (half a mile) higher than Everest, and it is gradually sinking. So Mauna Kea is as high as a mountain can be on our planet, and this absolute limit is set by the strength of our gravity.

The definition of weight can get a bit convoluted, and we mentioned that there are caveats to the rule of thumb that your weight on Earth is 9.81 times your mass. One problem is that the strength of Earth’s gravity varies slightly at every point on its surface. The most obvious effect is altitude; on the edge of the Fish River Canyon I would weigh slightly less than I would if I stood on the canyon floor. That’s because at the top of the canyon I am further from the centre of Earth than I would be at the bottom, so the gravitational pull I feel is weaker. Earth is also not uniformly dense – some areas of Earth’s surface and subsurface are made of more massive stuff than others, which also affects the local gravitational field. To complicate matters further, Earth is spinning, which means that you are accelerating when you stand on its surface, which means that the strength of gravity you feel changes in accord with the equivalence principle; this acceleration increases as you go towards the Equator, reducing the gravitational acceleration you feel there. Earth bulges out at the Equator because it is spinning, which weakens the gravitational pull there still further. The upshot of all this is that you weigh approximately 0.5 per cent less at the North and South Poles than you do at the Equator. The effects of the varying density of Earth’s subsurface and the presence of surface features on Earth’s gravitational field have been measured to extremely high precision and presented as a map known as the geoid

NASA

If you took Olympus Mons and stuck it on Earth…it would weigh around two and a half times as much as it does on Mars… A planet the size of ours cannot sustain a mountain of this size – it would weigh too much.

Towering over every other mountain in the Solar System is the extinct volcano, Olympus Mons. It is almost the height of three Mount Everests stacked on top of each other. The fact that a smaller planet has higher mountains is not coincidence; it is partly down to environmental and geological factors, but there is also a fundamental limit to the height of mountains on any given planet; the strength of its surface gravity. Mars has a gravitational pull at its surface of approximately 40 per cent of that on our planet.

ESA / DLR / FU BERLIN (G.NEUKUM) / SCIENCE PHOTO LIBRARY