# 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.

How long the duration was when the laws of physics became active after the big bang?And why the time is called plank's time?
how many galaxies in universe?
10 to the power 11
Wwe
testing
thnx
Shaikh
if u can count all the sands on the beaches and on the deserts then u will know how much it will be.......some said more than 1,000,000,000 galaxies are there in our universe...
Gospel
it's ok bro
Wwe
I hope you will become a member of nasa
Wwe
just new in this area, from art background not science
what's ur name wwe
Shaikh
The Observable Universe Contains Between 200 Billion To 2 Trillion Galaxies.....
are u sure?
Caleb
It actually hasn't been proven, people have just made estimations.
Ariana
the thing is u can't count,maybe the our number system will collapse there...who knows?
Abdullah
What's the exact time when the laws of physics became active after the big bang ?
Mostak
Actually we still don't know the exact number of galaxies in the universe, since the universe is bigger than we can imagine and it is still expanding even bigger today than yesterday
Ismaee
hi i need a telescope would you help mee
i can only advice you to go to some museum
Gospel
wher is it?
Nauman
but before that.......... i wanna know where are you from and from which state...and if you are nearer to that of the museum or will be able to go then.,.. i donot have any more words.
Gospel
What Do Plan To Observe?
sorry . i am from asia
Nauman
Check on amazon.Celeston telescopes are good for beginners.
priyanshu
Nauman
can you
Nauman
Asia...? from which country you belong into ...,,is it India or where?
Gospel
Do You Want A Telescope With ALot Of Maintenance Or Zero Maintenance?
pakistan
Nauman
with alot offf
Nauman
Any second hand online stores in your area? great for unused Christmas gifts etc..
Collins
So You Want A Newtonian Reflector?
So what i would put for that answer ?
The largest observatory complex in the world is on Mauna Kea, the tallest mountain on Earth. Whatare some factors astronomers consider when selecting an observatory site? Don’t forget practical ones.Should astronomers, for example, consider building an observatory on Denali (Mount McKinley) or MountEverest?
i need help on the the second part
Abdulla
Should astronomers for example considered building an observatory on Denali (Mount Mckinley) or MountEverest ?
Abdulla
I would take into consideration. climate weather pattern.
basim
also, could an establishment be built and be able to sustain years of being beaten by the weather?
basim
so far Mount Everest to my understanding is a difficult mountain
basim
it is because of light pollution in the cities that the observatories are situated in extremely remote areas.
Ashish
Everest ist the tallest mountain
Paul
how far us milky way galaxy?
oh man
We are part of the Milky Way Galaxy. We are within it, so it's not really away from us. Our galaxy is 100,000 light years in diameter. The nearest big galaxy is Andromeda, around 2 million lightyears away.
Jignesh
its 200,000 light years
Gospel
Where is Rice University located?
world
that couldn't be any more wrong 😂😂
jay
what even is this conversation😂😂!?
Herrera
Show Some Class.....
Houston, Texas.....
Houston
Preston
what is the first step to take as far as getting into the field of astronomy?
how are universe made up of?
saksi
lanika
I think that your question is- What is the universe made up of?
priyanshu
Ever heard of matter?
priyanshu
Well,the universe is made up of matter and possibly antimatter
priyanshu
Actually,it's made up of matter's smaller units,fundamental particles
priyanshu
But for normal purposes,say that it's made of matter
priyanshu
priyanshu
there are several theories,but the widely accepted one is the 'Big Bang Theory'.
priyanshu
In simple words,a veeeeeeery long time ago (20 billion years).
priyanshu
The universe was just inside a very small point.
priyanshu
Then,it started to expand.
priyanshu
It is still expanding today
priyanshu
But remember,it's just a theory,there is no hard and fast proof
priyanshu
oh d*mn😭😭😭😭 i cant do my astronomy quest
swagirl
up Down quarks atoms electrons proton neutrons
Blake
😭😭😭😭..
swagirl
it's come at Guran in soorat Alrahman that Allah say we made the all universe in 8 day.hope u will get your answer.
Tanvir
All Our Evidence Points To The Big Bang Theory, Like The Cosmic Microwave BackgGround, Left Over Radiation From The Big Bang.....
13.8 Billion Years Ago A Singularity Expanded.....
usually in astronomy quest there's many math formula. i'm difficult on them
practice!
Collins
?
swagirl
what was the question? if you need any help CrashCourse/astronomy and physics channel on YouTube are a good help.
Collins
yeah like... you know venn diagram?
swagirl
yeah it's for sets
Vinit
***youtu.be/uR70knMr2Hg
Collins
this might help you understand Venn diagram. and yes they're sets of numbers.
Collins
tx for u
swagirl
why the dark side of moon never face us? because moon dont rotate? why tho
im in 8th standard and my school teach us nothing about astronomy but i want to be an astronomer so i study from youtube and apps like this but sometimes i get language problems
Rishabh
dark**
Glory
side**
Glory
The Moon does actually rotate. It takes 27 days for the Moon to make a full rotation. Because it takes it so long and the Earth is rotating around the sun, it seems like the Moon is pretty much standing still. That's called "synchronous rotation".
steveh259
I ment to say "The Earth is orbiting around the sun"
steveh259
Tidal Locked.....
***svs.gsfc.nasa.gov/4442
Collins
thats cool
Gospel
ya gives you a look at how the moon orbits around us too and a perspective of how fast we are going. also gives the moon phases for the year! enjoy 😁
Collins
yes you are absolutely correct, it just like something like the milk of heaven
Subhasish
the far side of the moon is not the dark side of the moon.
Shawn
Tidal Locking?
priyanshu
what phases of the moon occurs when spring tide?
the moon has a 28 day cycle. you can download a moon calander or follow the pattern and make your own calendar. 1st full moon of the year 21/22nd Jan, its also a blood moon and a wolf moon. if this info helps at all. 😊
Collins
***svs.gsfc.nasa.gov/4442
Collins
How to determine the velocity of light?
using this formula c=1/√absolute permeability×absolute permittivity of free space i.e. c=1/√8.854×10^-12×4π×10^-7~3×10^8ms^-1
Vidyashankar
what is mean by Big Bang
What it means by Big Bang is how the world was made and came in to exist
Glory
The Big Bang Is The Best Explanation For What We See Around Us, And For The Origin Of The Universe, All Our Evidence Points To The Big Bang Theory..... And Was Proposed By A Belgian Priest.....
if the big bang theory is really true than i would like to know what exploded and where did it came from and where did the energy came from and where come from for the matter to expand into amd where did the organization come from and where did the information come from.........
Gospel
Some say that on the otherside of the singularity was a black hole. They say that at the other end of all black holes is another universe. Hence the multiverse. It's all theoretical at this point but less than a 100 years ago people believed that there was only 1 galaxy in the universe. So why
Rafael
should there be only 1 universe?
Rafael
universe.
Rafael
Big Bang is a theory on the origin of the universe.It says that the whole universe started as an infinitely dense and hot point,which expanded with time to form the universe in which we live today.
priyanshu
but if you study about string theory then you will come to know that before big bang there were string ( vibrating particles) which cause to form the conditions for big bang.
YOG
Hmm,actually,I dont know much about string theory,just a little.I just wanted to tell this guy what is the Big Bang Theory.
priyanshu
and according to string theory black holes are giant fuzzball of strings,there is no infinity dense singularity and no smooth horizon there is just only a ball of string
YOG
I think that is the problem. why don't we ask what was before the big bang. to explode there must be a reason or a change so if there is change there must be time also. science can't solve this . never . unless it accepts the existence of God . it is a matter of scientists will accept God. and every
KUMNEGER
For those arguing Religious Dogma over Science, regarding the Origins of the Universe, as well as the Origin of Man... Please pick up and read a Chemistry Book first, particularly in regard to Atomic Structure, Composition & the creation of other Elements from the Periodic Table... To start with.😉
Bill
Next you can look up the terms... Potential Difference, particularly associated with Electrical Charge or Energy & the Transfer of electrons from the outer Valence Shell of an Atom...and the production of ENERGY or Electrical Charge and Heat...😉😁😎
Bill
You should now begin to see the dots connecting themselves, from a scientific perspective, pretty quickly...😉😁😎
Bill
is astronomy a hard subject to learn because i want to be an astronomer
No it not it just takes time and effort to learn
Glory
Collins
I want to become a scientist but my knowledge is very low. How I can improve my knowledge?
Wwe
any form of science courses, there's free crash course channel on you tube. I've learned alot from this channel. covers most topics, science: biology chemistry physics, astronomy, ranges to anatomy, history and many many more. it's a never ending subject!
Collins
suggest a channel as a example
Wwe
CrashCourse is the name of the channel and there is a picture of an apple. look through their play lists. They will have other related channels they follow also.
Collins
thank you so much
Wwe
Collins
No actually astronomy is a very interesting subject and it's not so hard if you'll make research and read some books you'll find it so simple
Ismaee
to become a actually astronomer you need to learn physics and after that you can take job of astronomy
Sir
Which books should I read as a beginner?