Scaling Laws in Nature: The laws that explain the absence of a magnetic field on Mars and why squirrels can jump so high

We know that the interior of Mars is cooler than the interior of the Earth, but why is that the case? Both planets would have been very hot and fully molten during their formation 4.5 billion years ago, but Mars has cooled down faster than the Earth. One of the main reasons for this is that Mars has a smaller volume to surface area ratio than the Earth.

What determines an object’s heat energy and cooling rate?

The total heat energy in an object is determined by its temperature and mass. If we assume that the Earth and Mars started off with approximately the same temperature and density, then the energy contained within the planets only depends on their volumes. The rate at which an object loses heat (cooling power) is proportional to its surface area. The volume and surface area of a sphere with radius r is given by:

If we divide the volume (total energy) by the surface area (cooling power), the resulting ratio tells us how long it takes for the object to lose its heat. Let’s confirm this by dividing the units of energy with the units of cooling power:

Therefore, we can see that this ratio gives us the cooling time. Dividing the volume by the surface area shows us how the cooling time depends on the radius of the object:

The ratio tells us that the cooling time is proportional to r/3. This means that there is a linear relationship between the radius and cooling time. For example, a sphere that has a radius that’s 3 times larger will take 3 times longer to cool down. The radius of the Earth is 6371 km and the radius of Mars is 3390 km, which means that the Earth’s radius is about 1.9 times larger. If both planets had the same density and composition, the Earth would have taken about twice as long to cool down when compared to Mars. In reality, there are other factors, such as the decay of radioactive elements, that affect heat production and loss, but this geometric approach provides a reasonable estimate.

This dependence of heat loss on size also applies to living creatures. The heat produced by an animal depends on its volume, while the cooling depends on its surface area. If you shrink an animal, its volume (and heat production) will decrease faster than its surface area (and cooling rate). Hence, smaller animals will have a shorter cooling time, and they tend to have a faster breathing and heart rate to generate and distribute heat in their bodies at a faster pace.

The lack of a global magnetic field on Mars

The interior of the Earth has a large quantity of molten metal, and the temperature is highest at the centre. Since the interior heat is being lost to outer space through the Earth’s surface, the temperature gradually drops as we move away from the Earth’s centre, and the coldest region is near the surface. Hot fluids are less dense than colder fluids and rise to the top. This means that the hot molten metal from deep inside the planet rises up towards the surface, where it cools down and drops back down towards to centre where it is heated again. This cycle repeats and creates a continuous convection cycle. The flow of the metal within the Earth creates the planet’s magnetic field and protects us from solar radiation.

Since the interior of Mars has already cooled down a lot, it has a very thick crust. The average thickness of the crust on Mars is about 50 km, while the average is 24 km for Earth. The thick crust on Mars acts as an insulator and reduces the heat loss from the interior, similar to wearing a planet-sized sweater! This means that the top of the molten layer is at a similar temperature as the centre of Mars. Consequently, the temperature difference is a lot smaller within Mars. As a result, there is no strong convection movement within Mars as the molten metal moves very slowly. Hence, the magnetic field of Mars eventually switched off due to the planet’s rapid cooling. The 2003 science-fiction movie, The Core, attempted to prevent the Earth’s magnetic field from shutting down. Without a magnetic field, the atmosphere of Mars was gradually stripped away by the solar wind, which is a stream of high-energy charged particles from the Sun. The loss of most of the atmosphere meant that the liquid water on Mars then quickly evaporated and was lost into space.

Scaling laws of living creatures

The volume to surface area ratio is also important for the physiology of living creatures. If we assume that all creatures have approximately the same density, then their weight mainly depends on their volume. Their muscle strength, on the other hand, depends on the cross-sectional area of the muscles. For example, a person’s ability to lift heavy weights depends on the cross-sectional area of their arm muscles.

We know that volume depends on the length-cubed, and the area depends on the length-squared. This means that if the size of a creature is doubled, then its volume (and weight) increases by a factor of 8, while the cross-sectional area (and muscle strength) increases only by a factor of 4. Therefore, large creatures (like elephants) have thicker limbs to support their weight when compared to smaller creatures like lions. If we enlarged a lion to be as large and heavy as an elephant, it would not be able to support itself, as its weight would have increased a lot more than its leg muscle strength.

Jumping out of a blender

One of the job interview questions at Google asked the applicant something along the lines of this: You are shrunk down to the size of a coin and dropped into a blender. The lid is open and you need to escape before the blender is turned on. What’s the best method of escaping?

Turns out that the best way to get out of the blender is to just jump out! Have you noticed how some large animals like horses and small animals like squirrels can jump to the same height? This is because small animals like squirrels have a higher strength-to-weight ratio than most larger animals. This is the reverse of the scenario mentioned earlier. When you shrink an animal, its weight reduces faster than its muscle strength. As a result, a lot of smaller animals can jump to many times their own height, which is something that large animals cannot do. So, if you are shrunk to the size of a coin, your legs muscles will be a lot more powerful with respect to your (greatly reduced) bodyweight. You can then easily jump out of the top of the blender like a squirrel!

What if gravity was stronger?

There are planets elsewhere in the universe which are similar to Earth but have a much higher mass (“super-Earths”). This means that the gravitational force on their surface will be much higher. If animals had evolved on planets like these, their weights would be higher but their muscle strengths would be unaffected as it depends on the electromagnetic force within the muscle tissue and not on the planet’s gravity. Hence, they would need to be smaller and/or have thicker limbs to support their weight.

Another possibility to consider is if gravity was intrinsically stronger in our universe. Gravity is considered the weakest of the four fundamental forces (the others being the electromagnetic, strong, and weak nuclear forces), but what if it was twice as strong as it is currently? This would result in the gravitational force of every object being twice its current value. Doubling the gravity of the Earth would also result in animals being smaller and/or having thicker limbs for the same reason as mentioned above. Moreover, the orbits and properties of all celestial bodies would change drastically! What would happen if we instead doubled the strength of the electromagnetic force, which is the main force involved in muscle strength? Thinking about the effects of modifying fundamental aspects of nature is a highly interesting endeavour, as it can have profound and unexpected consequences on the functioning of the universe…