Will the Moon Fall Down on Earth due to their Mutual Gravitational Attraction as someone asked?

I received this comment and question from someone in my WhatsApp chit-chat group.

It reads as quoted:

“Dear Professor Lim Ju Boo

 

I read with great interest your scholarly detail explanation about the varying distances between the earth and the sun, and whether it affects the temperature of our earth

Also, your very interesting account how far the moon ought to be to cover the entire, half and one quarter of the sky following the false report about the "supermoon" except when seen over the horizon as you explained 

This leads me wondering if the moon can be attracted by the earth's greater gravity until a really super large moon coming close to us one day

Can you tell us if there would be this possible in a simple matter without using much mathematics which would be beyond most of us. I know you did a postdoctoral in astronomy at Oxford but this would be out of reach by most of us. 

Thanking you in advance”

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Here’s my reply to him:

Thank you for your question and curiosity. Let me first try to answer your question in a non-mathematical way as you requested before proving it mathematically as you wanted to avoid. Without facts and figures from mathematics you may think I am putting up some unsubstantiated theory or just a claim.

We shall try the descriptive, non-mathematically approach first so that it will be reachable to the general readers.

The non-mathematically analogy is like placing a stone or a projectile representing the moon in a sling with the string representing the gravity between the stone or a projectile, and the hand as the earth holding the sling at the other end.

When he swings the sling round and round it represents the orbit of the Moon around the Earth.

If the sling is swung round and round, the stone is held by the string. It does not fly off, nor does it get attracted or ‘fall down’ towards the hand.

But if he releases the string, the stone flies off at a tangent to the circular path since the string represents the pull of gravity between the earth and the moon.

The tendency for the stone to fly outwards away from the centre is called the centrifugal force, while the inner force held by the string is called the centripetal force to allow an object to follow a curved path. Both these two forces must just balance each other in order for the stone to stay in a circular motion.

In the same way, the Moon is being held tightly by the mutual Earth-Moon gravity as she takes her orbit around the Earth, if not the Moon would fly away like a projectile in a slingshot.

Hope this explains in a very simple non-mathematically way as you wanted.

Now, let’s try to be a bit more technical to prove this concept.      

The gravitational force (F) of attraction between the Earth and the Moon is governed by Isaac Newton’s equation on gravity:

 F = G (m1 x m2) / d2 

Where, G is the gravitational constant (6.6743 ± 0.00015) × 10^−11 m3 kg−1 s−2

m1 is the mass of the Earth (5.972 x ^24 kg)

m 2 is the mass of the Moon (7.34767309 x ^22 kg)

d, is the distance between Earth and Moon (384,400 km = 3.844 x ^8 meters)

In order to calculate the force of attraction (F) between these two heavenly bodies, we apply the above Newton’s equation:

F = G. m1m2 / d2 = 6.674×10^ – 11 x 5.972×10^24 x 7.34767309×10^22 / (3.844×10^8)^2 

= 2.928571469 x 10 ^37 / (3.844×10^8)^2 

= 1.98 x 10 ^20 (two hundred million trillion) Newton

Let us now consider if the outwards centrifugal force of the Moon is the same as the inner gravitational force that alllow the Moon to orbit around the Earth? The centrifugal force basically uses the same formula as for centripetal force.

As the moon goes around Earth every 27.3 days or 2,358,720 seconds (T) at a distance of r (3.85 x 10^8 metres) from Earth, we need to know if both the gravitational and centrifugal forces (F) balance each other?

In order to do that, we need to know the velocity (V) of the Moon around the Earth first. This would be:

V = 2 pi r / T = 1025.567 metres per second

If the moon’s mass is 7.35 x 10^22 kg, the force of the Earth’s gravity exerts on her as it orbits Mother Earth would be:

F = mv2 / r = (7.34767309 x 10 ^22 kg) . (1025.567^ 2) / 3.85 x 10^8 m.

= 7.72819197 x 10 ^28 / 3.85 x 10^8 metres

= 2.0 x 10^ 20 Newton (two hundred million trillion Newton)

where m = mass of the Moon (7.35 x 10^22 kg)

V = Moon’s velocity of orbit (1025.567 metres per second)

r = Earth-Moon distance (3.85 x 10^8 metres)

Hence, we can see the centrifugal or outward force of 2.0 x 10^20 Newton of the Moon trying to escape Earth’s gravity is exactly the same as the inner gravitation force that keeps the Moon in orbit. 

The outward centrifugal force away from the Earth's centre and inner gravitational force  exactly balances each other to bind the Moon in a circular motion around her Mother Earth.

Hence the Moon can never fall down to Earth

I hope we have proved this concept non-mathematically, and have verified it mathematically in answering your question and curiosity

 Quod Erat Demonstrandum (Q.E.D)

 

Having said that, on the contrary, the Moon is actually receding away from Earth due to the tides she exerts on Earth. How this happens is because the Earth's rotation is essentially slowing down by around 6 hours over the past 2740 years. This causes the period of a 24-hour day to increase by about 1.78 milliseconds over the period of a century. This is about 0.0015 seconds per century, or about 0.000015 seconds each year. This means 600 million years ago a day was only 21 hours.

 The slowing down is due to loss of rotational energy from friction caused by tides on the oceans and land mass. The tides are due to the gravitational pull of the Moon on Earth, both over the oceans and land mass. The ocean tides and solid tides over the land, albeit hardly noticeable over land, sweep across them especially over the oceans twice a day.

The tides over the oceans and seas causes the waters to bulge up on opposite sides of the Earth. This causes the Earth to lose rotational energy and angular momentum which currently is about 7.2 x 10 ^33 kg m^2 s^-1

Angular momentum is a closed system that cannot be created or destroyed, but is being conserved. In other words, if the Earth loses angular momentum, the Moon must gain it. In order to conserve angular momentum, it causes the Moon to recede from Earth at a rate of 3.78 cm per year. This drift has been going on for 4.5 billion years

Computer simulations showed that at the time of the formation of the Earth-Moon system 4.5 billion years ago, the Moon was much closer to the Earth. It was then a mere 22,500 km away.

The current average distance between Earth and the Moon is about 384,400 km or 3.844 x^ 8 meters.

During the formation of the Moon, the apparent size of the Moon in the ancient skies was about 8.7 degrees. That was 16.7 times larger than seen today. The current seeming size of the Moon is 0.52 degree

However, the Moon may not continue to drift away from Earth forever. A time shall come, about 50 billion years from now when the Moon’s orbit will reach its farthest. By then, the length of a lunar orbit around the Earth will be about 47 days instead of the current 27.3 days.

By that time, the rotation period of Earth will also be 47 days. This implies that one side of the Earth will forever face the Moon just as one side of the Moon is currently, and permanently facing the Earth. Both Earth and Moon will be gravitationally locked to face each other on one side.

In such an event when the Earth and Moon are locked to face each other on only one side, they will no longer have any tidal effects on each other, and the Moon will also stop drifting away.

But by then, human civilization will no longer exists, and probably all life on Earth will be extinct

The Sun itself will not even last that long. It has only about 5 billion  more years to go before it uses up its hydrogen fuel by fusing about 564 million tons hydrogen per second into 559.7 million tons of helium, losing a mass of about 4.3 million tons per second. The remaining 4 million tons is converted into energy

When the Sun uses up her fuel, she will bloat up into a red giant to engulf Mercury, Venus, Earth and possibly beyond  

I think I have answered and explained more than requested. We need to stop here lest we are tempted to write another chapter on astronomy.

Thank you for reading

jb lim