# 3.3 Newton’s universal law of gravitation  (Page 3/4)

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Newton’s law also implies that gravity never becomes zero. It quickly gets weaker with distance, but it continues to act to some degree no matter how far away you get. The pull of the Sun is stronger at Mercury than at Pluto, but it can be felt far beyond Pluto, where astronomers have good evidence that it continuously makes enormous numbers of smaller icy bodies move around huge orbits. And the Sun’s gravitational pull joins with the pull of billions of others stars to create the gravitational pull of our Milky Way Galaxy. That force, in turn, can make other smaller galaxies orbit around the Milky Way, and so on.

Why is it then, you may ask, that the astronauts aboard the Space Shuttle appear to have no gravitational forces acting on them when we see images on television of the astronauts and objects floating in the spacecraft? After all, the astronauts in the shuttle are only a few hundred kilometers above the surface of Earth, which is not a significant distance compared to the size of Earth, so gravity is certainly not a great deal weaker that much farther away. The astronauts feel “weightless” (meaning that they don’t feel the gravitational force acting on them) for the same reason that passengers in an elevator whose cable has broken or in an airplane whose engines no longer work feel weightless: they are falling ( [link] ). In the film Apollo 13 , the scenes in which the astronauts were “weightless” were actually filmed in a falling airplane. As you might imagine, the plane fell for only short periods before the engines engaged again.

When falling , they are in free fall and accelerate at the same rate as everything around them, including their spacecraft or a camera with which they are taking photographs of Earth. When doing so, astronauts experience no additional forces and therefore feel “weightless.” Unlike the falling elevator passengers, however, the astronauts are falling around Earth, not to Earth; as a result they will continue to fall and are said to be “in orbit” around Earth (see the next section for more about orbits).

## Orbital motion and mass

Kepler’s laws describe the orbits of the objects whose motions are described by Newton’s laws of motion and the law of gravity. Knowing that gravity is the force that attracts planets toward the Sun, however, allowed Newton to rethink Kepler’s third law. Recall that Kepler had found a relationship between the orbital period of a planet’s revolution and its distance from the Sun. But Newton’s formulation introduces the additional factor of the masses of the Sun ( M 1 ) and the planet ( M 2 ), both expressed in units of the Sun’s mass. Newton’s universal law of gravitation can be used to show mathematically that this relationship is actually

${a}^{3}=\left({M}_{1}+{M}_{2}\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{P}^{2}$

where a is the semimajor axis and P is the orbital period.

How did Kepler miss this factor? In units of the Sun’s mass, the mass of the Sun is 1, and in units of the Sun’s mass, the mass of a typical planet is a negligibly small factor. This means that the sum of the Sun’s mass and a planet’s mass, ( M 1 + M 2 ), is very, very close to 1. This makes Newton’s formula appear almost the same as Kepler’s; the tiny mass of the planets compared to the Sun is the reason that Kepler did not realize that both masses had to be included in the calculation. There are many situations in astronomy, however, in which we do need to include the two mass terms—for example, when two stars or two galaxies orbit each other.

Including the mass term allows us to use this formula in a new way. If we can measure the motions (distances and orbital periods) of objects acting under their mutual gravity, then the formula will permit us to deduce their masses. For example, we can calculate the mass of the Sun by using the distances and orbital periods of the planets, or the mass of Jupiter by noting the motions of its moons.

Indeed, Newton’s reformulation of Kepler’s third law is one of the most powerful concepts in astronomy. Our ability to deduce the masses of objects from their motions is key to understanding the nature and evolution of many astronomical bodies. We will use this law repeatedly throughout this text in calculations that range from the orbits of comets to the interactions of galaxies.

## Calculating the effects of gravity

A planet like Earth is found orbiting its star at a distance of 1 AU in 0.71 Earth-year. Can you use Newton’s version of Kepler’s third law to find the mass of the star? (Remember that compared to the mass of a star, the mass of an earthlike planet can be considered negligible.)

## Solution

In the formula a 3 = ( M 1 + M 2 ) × P 2 , the factor M 1 + M 2 would now be approximately equal to M 1 (the mass of the star), since the planet’s mass is so small by comparison. Then the formula becomes a 3 = M 1 × P 2 , and we can solve for M 1 :

${M}_{1}=\frac{{a}^{3}}{{P}^{2}}$

Since a = 1, a 3 = 1, so

${M}_{1}=\frac{1}{{P}^{2}}=\frac{1}{{0.71}^{2}}=\frac{1}{0.5}=2$

So the mass of the star is twice the mass of our Sun. (Remember that this way of expressing the law has units in terms of Earth and the Sun, so masses are expressed in units of the mass of our Sun.)

Suppose a star with twice the mass of our Sun had an earthlike planet that took 4 years to orbit the star. At what distance (semimajor axis) would this planet orbit its star?

Again, we can neglect the mass of the planet. So M 1 = 2 and P = 4 years. The formula is a 3 = M 1 × P 2 , so a 3 = 2 × 4 2 = 2 × 16 = 32. So a is the cube root of 32. To find this, you can just ask Google, “What is the cube root of 32?” and get the answer 3.2 AU.

## Key concepts and summary

Gravity, the attractive force between all masses, is what keeps the planets in orbit. Newton’s universal law of gravitation relates the gravitational force to mass and distance:

${F}_{\text{gravity}}=G\frac{{M}_{1}{M}_{2}}{{R}^{{}_{2}}}$

The force of gravity is what gives us our sense of weight. Unlike mass, which is constant, weight can vary depending on the force of gravity (or acceleration) you feel. When Kepler’s laws are reexamined in the light of Newton’s gravitational law, it becomes clear that the masses of both objects are important for the third law, which becomes a 3 = ( M 1 + M 2 ) × P 2 . Mutual gravitational effects permit us to calculate the masses of astronomical objects, from comets to galaxies.

what is comets , astroids ,
hi, I am yamini, I am in class 8 but very much interested in astronomy and go to NASA, what are the subjects in which I can master and lead to NASA.
Hello Yamini, Im Ken and I'm avery intrested in joining the NASA too. Is nice to meet you.
Ken
very*
Ken
nice to meet u ken
yamini
What I know about the subjects is that you have to be a master on science and math, also if you know about aviation is better too
Ken
I read that the Russian language is very important, is not a requirement but it's like an extra point!
Ken
yeah like physics, chemistry and maths, they are my most favorite.
yamini
than you can go free of cost
Arush
Yes, That's right!
Ken
how
yamini
Oh! Free of cost?
Ken
hello yamini nice to meet you
Burak
what is time
Time is relative
mrunal
pls elaborate
sakshi
the clear defination.I know that.
Abdul
In planet mars there the life exits or not and is there water there
see till now nothing can be found as u know that the curiosity rover has struck in mars
Maya
It has been proven that there are water molecules on Mars but not enough that most lifeforms could thrive upon.
Ariana
There are huge amounts of water in the ice caps and under the surface. The surface and chemistry indicate that Mars had cosiderable amounts of water on its surface in the past.
Julius
i think in some ways vedas are also correct but not everytime
Maya
I agree
sakshi
hmm even I agree
Samuel
Is there any patened theory about time relativitg in growth and development?
some astronomer's says that there is no alien exist but why search for extra terrestrial intelligence center is established
No One Knows That For Absolute Fact, The Universe Is Too Huge To Have Any Type Of Idea About What Exist In The Far Reaches Of Our Universe.....
Check Out The Drake Equation.....
their should be aliens as like ours there would be another planet
Maya
which could have existed life on it
Maya
Maya
can kepler 1st law be applied on all the planets of the universe
Maya
hello, anyone home?
Denise
sjskskfhjkkktewqqw and try?
what is this ?
Samuel
hi I am Samuel from India mumbai
Samuel
nice to meet you
Samuel
thats my question, what is this?
penzias and wilson's a discovery of the cosmic microwave background is a nice example of scientific serendipity-something that is found by chance but turns out to have a positive outcome
how should I make my carrier in astronomy
I think that Newton's third law is not appropriate if any also thinks like this please reply me
Ayush
Huh
why u think so
joseph
yes Ayush u are right
Yoganshu
I think when we apply force to a object it start moving but , a/c to Newton's third law every action has equal and opposite reaction,so object should also exert equal force on us and it should not move due to balanced force
Ayush
if I am not right then reply me
Ayush
no
Zack
because of friction that opposes that force and help us to move ahead
Manish
but this is not satisfied as third law say another thing
Ayush
you are telling why object moves
Ayush
you have to think a/c to third law
Ayush
its because of its mass
Maya
because it is applying equal and opposite force but also our mass is also less in comparison to the object
Maya
which is why we cant move the object but it can make move us
Maya
manish is too correct in his place because we need to apply force which would overcome the frictional force
Maya
My dear friends, can u plz tell me that among u guys who are in the field of cosmology
😢I am not there in cosmology
Samuel
Just A Science Fan.....
Samuel
I am also not in cosmology but I am just a fan or we can say science and part of NASA is my dream
Yoganshu
yoganshu Arya same here
Samuel
you are from which country
Yoganshu
hi yoganshu
Samuel
India
Samuel
which state
Samuel
I am also from India
Yoganshu
from delhi
Yoganshu
and u...?
Yoganshu
I am from Maharashtra
Samuel
from which state?
Yoganshu
You are a ASTRONOMER ...
Yoganshu
or a scientist..
Yoganshu
or just a member
Yoganshu
What is time...? not about Newton= time is constant..... that all scientists openions n point of view I m knowing. . what can be the Perfect Definition of Time
Ayush
Who is the best astronomer of India at present time
Jayant Narlikar, Proponent Of Steady State Cosmology.....
What is the real colour of sun rays
white.. so white it becomes violet.. so violet it become ultraviolet
Tom
white and red and yellow
Bianca
Vibgyor
Samuel
the real colour of sunlight is White
the Sun's has a variety of waves all throughout the elextromagnetic spectrum.
Jacie
we only see it as a few bc of how some of them get redshifted (? can that term be applied for something so local?) by some particles in our upper atmosphere
Jacie
Vibgyor will be when, the white light will pass through the clouds ( prism ) then Refraction phenomenon leads us to 7 colours splitting from a single colour "White " light
so the sun rays r of White colour
Taurus in astronomy and horoscope?