It all gets a bit hairy, but we'll do best to make sense of it.
We'll look at motion and orbits again when we do a further discussion of Newtonian physics and gravity in IAL 5: Newtonian Physics, Gravity, Orbits, Energy, Tides.
A reference frame is just a coordinate system (or some simpler kind of reference system if you want to think that way) that one lays on all space or some subset of all space.
It allows you to locate objects in space and describe their motion (in physics jargon, their kinematics).
Usually, there is a specified origin for the reference frame.
There are infinitely many reference frames that one can use: they can have different origins, different kinds of coordinates (e.g., Cartesian coordinates, spherical coordinates), and be in any kind of motion.
See the figure below for an example reference frame which includes a time coordinate.
Reference frames
can be just geometrical descriptions or they can be
attached to some physical object, and/or have a definite physical
nature which is the case for
inertial frames
most imporantly of all.
Inertial frames
are reference frames
which have physical nature---they are NOT arbitrary human choices.
The modern
inertial frame
is defined in
general relativity
discovered by Albert Einstein (1879--1955).
General relativity
is our best theory
of gravity
and the main theory ingredient
of cosmology.
In fact,
free-fall frame is virtually
a synonym
for the modern definition of an
inertial frame.
Yours truly often uses either term to mean
inertial frame.
Note the definition is for the ideal
inertial frame.
Real
inertial frames
are almost always only approximately
inertial frame
though in many cases they approach
ideal inertial frames
so closely that the difference is negligible.
Some further explication is needed after the above figure which just recapitulates and expands a bit.
We should call them free-fall frames
since that corresponds to our modern understanding.
And yours truly's
finds inertial frame
hard to pronounce
and yours truly when speaking is always confusing it with its opposite
the non-inertial frame.
One might suggest:
non-inertial frame →
geometrical frame
since physical laws are NOT
referenced to this kind of reference frame
but, of course, they are "physical frames" in another sense.
But the dead hand of tradition has stuck us with
inertial frames
and non-inertial frames.
The common symbol is "g" put in vector notation: e.g.,
boldface vector notation g or, as yours truly often writes it in
HTML, the vector notation vec g.
The gravitational force
on a body in a small compared to
variations in vec g is just
This is because the UNIFORM EXTERNAL
gravitational field
pulls on every bit of a system equally.
The external gravitational field
of the definition just has to be uniform enough for your purposes or
the non-uniformities have to be correctable in your calculations.
So, in fact, exact
inertial frames are also
idealizations, but in many cases approached to high
accuracy/precision.
Some
reference frames are more
inertial than others:
there are varying degrees of approximation to exact
inertiality.
The center of mass
is a mass-weighted average position for a
system.
We explicate
center of mass
further below in
subsection Center of Mass.
The center of mass
follows the exact
free fall trajectory.
The reference frame
with this origin
CANNOT be taken to rotate with respect to the
free-fall trajectory
because that would be an
acceleration.
An acceleration is
change in speed
and/or direction.
We explicate
acceleration
further below in subsection
Acceleration, Force, and Inertial Frames.
Let's call the
inertial frame
with the origin
at the center of mass,
the center-of-mass
inertial frame.
It is an non-inertial frame
for that region.
To further explicate inertial frames,
we need to define what we mean by
acceleration
and
force.
An acceleration
is a change in speed AND/OR a change in direction.
These two kinds of change are illustrated in the two figures just below.
To explicate:
Often the center-of-mass
inertial frame.
A force can depend on
mass (gravity),
electric charge
(the electromagnetic force),
relative position (gravity,
the electromagnetic force),
velocity
(the magnetic force),
and other things.
Note that Newton's laws of motion
are referenced to inertial frames.
It is just part of their statements---though this part is often omitted
in initial presentations of the laws to students.
What "referenced to" means is that the laws do NOT work if NOT applied relative to
inertial frames.
This does NOT mean the physical laws
are wrong somehow since they are explicitly or implicitly formulated as
referenced to inertial frames.
General relativity
is a great exception to the rule that
physical laws are referenced
to inertial frames.
General relativity is the
other way around.
It tells us what
inertial frames are and where they
are.
See subsection
Inertial Frames above.
Probably, the
2nd law of thermodynamics
is also NOT necessarily referenced to
inertial frames.
Yours truly needs to ponder this fine point.
Well you could always just switch
from
a non-inertial frame
to an inertial frame.
They are just
frames of reference
after all.
But sometimes that's NOT convenient.
For example, if you are embedded deeply in a
rotating reference frame,
that is your natural
reference frame for
most purposes.
The trick is then to introduce
inertial forces
which are NOT real
forces,
but just force-like
quantities in the physical
formulae that
give the effects
of being a
non-inertial frame.
We discuss two inertial forces
below in
subsection
Inertial Forces on the Earth's Surface.
We usually treat the surface of the Earth
as an inertial frame.
Newtonian physics would NOT
be much use in everyday life if
we could NOT do so.
All reference frames
NOT accelerated with respect to the
ground also serve
pretty well as
inertial frames.
See a very pretty pretty piece of the surface of the Earth
in the figure below.
Well spacecraft
Earth is in
free fall.
The center of mass (CM)
of Earth
is in orbit
in the external
gravitational field
of the Sun,
Moon,
and, to a much lesser degree, other
Solar System
astro-bodies.
There are two complications:
We call this tidal force.
It's a stretching force
that is very weak over short distances.
So it doesn't stretch you and me, but it stretches the
World Ocean
to give us the tides.
See the figure below.
Therefore the surface of the Earth
CANNOT be exactly an
inertial frame
but for most, but NOT all, purposes, it's
approximately an
inertial frame
It is inertial enough.
Non-inertial frame
effects can be treated, as discussed in the subsection just above, as
inertial forces
which is just formalism for treating these
non-inertial frame
effects and NOT real
forces.
We explicate the
inertial forces
on the surface of the Earth
a bit more in the subsection
Inertial Forces on the Earth's Surface
given just below.
There are two main
inertial forces
on the surface of the Earth:
The centrifugal force is
NOT a real force.
It's
Newton's 1st law of motion
in action.
You are trying to go in a straight line
and need to exert a real force to keep
in rotation.
Effectively,
the centrifugal force is
an outward "force" from a center of rotation
in the rotation's own
rotating reference frame.
The centrifugal force
is the thing that tries to throw you off
playground merry-go-rounds:
see the figure below.
The centrifugal force
of the Earth
is zero at the poles
where there is no rotation
and strongest at the
equator
where the velocity
of rotation is
0.4651 km/s relative to the
CM
free-fall frame
of the Earth.
The centrifugal force
effect on the Earth's gravity is below
human perception,
but is quite measurable:
e.g., with a gravimeter.
Given the high velocity
at the equator
equator compared
to playground merry-go-rounds,
you may wonder the
centrifugal force
of the Earth
is so small.
The essential answer is from the
angular velocity of the
Earth: 360° per day.
You wouldn't notice any
centrifugal force
on playground merry-go-rounds
either if it were going that slowly.
To be more physicsy, the
centrifugal force per unit
ranges from 0 at the poles
to ∼ 0.05 N/kg (m/s**2) at the
equator
which causes
Earth's effective gravitational field
vary from ∼ 9.83 N/kg at the
the poles
to ∼ 9.78 N/kg at the
equator
(see Wikipedia: Gravity of Earth: Latitude).
This means you weigh 0.5 % less at the
equator than at the
poles---measurable, but
noticeable to
human perception.
There are also small variations in the
Earth's gravitational field
due to elevation
and varying geology.
All these variations are easily measured too, but are below
human perception.
The other inertial force
on the surface of the Earth
is the Earth's
Coriolis effect (AKA Coriolis force).
It is an effect due to motion in a
rotating reference frames.
For the Earth,
it's NOT noticeable on small scales, but it gives rise to
the vortex motion
of cyclones
(see Wikipedia: Cyclone: Structure)
and anticyclones
(see Wikipedia: Anticyclone: Strike).
The Coriolis effect
on Earth and
other planets is
explained in the figure below.
What the heck is center of mass?
Don't panic,
we'll
NEVER calculate a center of mass---we
just need to grok the
concept.
The figure below illustrates and explicates
center of mass.
There is no place else they could be given that they are mass-weighted average positions.
One can find the centers of mass
in the figure below
by inspection.
However, there is a simple
empirical method for
finding the
center of mass for
rigid systems.
The method is illustrated in the figure below.
Much of the analysis of motion from
Newtonian physics
requires center of mass.
But to give a specific example, we need
center of mass
in understanding how things are held static from a free pivot point: e.g.,
for hanging objects or balancing them.
To explicate:
Why do WE (i.e., YOU) need the
concept of center of mass
for astronomy and an understanding
inertial frames
in astronomy?
In astronomy
Gravitationally-bound systems,
make up most of the astrophysical realm.
And the component
astro-bodies of a
Gravitationally-bound system
orbit
their mutual
center of mass.
So obviously
center of mass
is vital for astrophysics.
Why does the center of mass
play the role it does? Short answer:
because Newtonian physics
(i.e.,
Newton's 3 laws of motion
plus
Newton's law of universal gravitation)
so dictate.
For Gravitationally-bound systems
the center of mass
is also called
the barycenter.
Below is a figure illustrating a
gravitationally-bound system
of astronomical objects
orbiting the system
barycenter
(i.e., center of mass).
The Sun's mass
is 99.86 % of
the Solar System
mass
(see Wikipedia: Solar System:
Structure and composition).
This overwhelming dominance of the
Sun's mass
means that to good approximation the
Sun's center is
approximately the
barycenter
(i.e., orbital system
center of mass)
of Solar System
and can be used to define an approximate
Solar-System
inertial frame
(i.e., free-fall frame)
which is good enough for many purposes.
The fixed stars are
just the relatively nearby stars
(e.g., those that historically define the
constellations)
that are moving in very similar
orbits
to the Sun's
orbit
around the Milky Way.
The frame of reference
defined by the fixed stars
(an average frame or the frame any one individually)
is technically NOT quite as an approximation to
an exact inertial frame
as that defined by the
Solar-System
barycenter.
The fixed stars are all in their
own free-fall frames
which are slightly different than that of the
Solar-System.
barycenter
inertial frame
But for most purposes, the distinction between
the Solar-System
barycenter
inertial frame
and
the frame of reference
of the fixed stars is vanishingly
small.
And for many purposes using the
frame of reference
fixed stars is easier to use
since one can easily make measurement with resprect ot the
fixed stars
and historically this was the normal procedure.
In fact, we often reference motion to
the fixed stars as a way of
meaning relatively to an exact local
inertial frame
for Solar System.
The gravitational force
the Sun is the
main determinant of the
structure of the Solar System.
It pulls the planets
into their
orbits---which means the
planets are in states
of acceleration.
Recall
Newton's 3rd law:
for every force there is an equal and opposite
force---but note these two
forces do NOT have to be on
the same body, and so
just do NOT just cancel out all the time.
Thus the
planets exert equal
gravitational forces on the
Sun to what the
Sun exerts on them.
So the
Sun should also be
accelerated in the frame of fixed stars.
But, as we will discuss
IAL 5: Newtonian Physics, Gravity, Orbits, Energy, Tides,
acceleration
is proportional to the force
and INVERSELY proportional to
mass.
So the gravitational forces of the
planets
on Sun,
do NOT affect the motion of the Sun very much.
Note to 1st order
only the Sun's
gravitational force
affects a planet.
Thus, to 1st order
the Sun
and each planet form
gavitational two-body system:
i.e., a
system consisting of only 2
gravitationally interacting bodies.
If the
Sun
suddenly disappeared, the planets
would fly away from each other in space
and never meet again because
the major source of gravity was gone:
gravity is proportional to mass.
The instructor can---if he remembers---do a demonstration with a swirling object.
In gravitationally-bound
gavitational two-body system
(exact or approximate),
the two bodies orbit
their mutual
center of mass
(or barycenter)
in elliptical orbits.
The center of mass
defines the local
inertial frame.
If one body is much more massive than the other,
it is approximately at the center of mass
and in this case we say that
the less-massive body orbits the more-massive body
in an inertial frame.
As a shorthand,
we say that the
less-massive body "physically" orbits the more-massive body
in order NOT to always have
to mention
inertial frame all the time.
Or for an even shorter shorthand,
the less-massive body orbits the more-massive body.
This is whay we say the
planets
orbit
the Sun
and the moons
orbit
their parent planets.
Actually, the external
gravitational field
acting on
the Earth-Moon system
has a strong variation
(i.e, a tidal force)
and this CANNOT be neglected in
calculating the
Moon's orbit.
Isaac Newton (1643--1727)
once said that trying to calculate the
Moon's orbit was the
one problem that made his head ache.????
We have often do take a geometrical perspective for observations
and say that the Sun
orbits
the Earth
and in fact that everything orbits
the static Earth.
But we understand what we mean when we take the geometrical perspective
and often say from the
perspective of the Earth
for observational work.
This has been a long story to explain why we say
"the planets orbit the Sun".
We've used the
CM
free-fall frame
of the Earth
as an example
CM
free-fall frame.
It's the one we use to analyze motion on the surface
of the Earth.
However,
free-fall frames
are everywhere in the
astrophysics realm.
***under construction below***
Other astonomical examples, are
the CM
free-fall frames
of the
Solar System
(which is nearly the center of the
Sun and is to good approximation
the reference frame
of the fixed stars),
other planetary systems,
star clusters,
galaxies,
and galaxy clusters.
All of these are gravitationally bound
systems,
and so are natural
CM
free-fall frames
for observable universe.
Notice there is a NESTED HIERARCHY of
CM
free-fall frames
with lots of rotation:
planetary systems
are usually in galaxies which are often
in galaxy clusters.
See the figure below.
Dense astronomical objects
(e.g., planets and
stars) are held up
against self-gravity
by the pressure force.
Actually, stars would be held
up by the kinetic energy
of their particles even if
the particles did NOT
collide to create the
pressure force.
Stars would have very different
structure if collisions were turned off---they might be something like
dark matter halos in structure.
The factoid
stars would be
kinetic energy
of their particles
makes the virial theorem
useful for analyzing
stars.
In the
comoving frames of the expanding universe.
See subsection
Comoving Frames below.
Say you have a large system
with a center of mass (CM)
in free fall
relative to an external
gravitational field.
The center of mass
of the Earth
is to accuracy/precision
the geometric center of the Earth.
So we can write
Newton's 2nd law of motion (AKA F=ma)
for a general body
Now we expand as follows:
Using the above expansion,
Newton's 2nd law
can be written
If we chose our
local reference frame
really well, the term m*(vec Δg_e - vec a_f) is small and can be neglected.
To pick up our example of Earth.
This is what we do for the surface of the
Earth is
usual our
local reference frame
and it is
a good approximate
inertial frame
for most purposes since the m*(vec Δg_e - vec a_f) can be neglected sort of.
Note the surface of the
Earth
is NOT a
free-fall frame
relative to the Earth.
But it is approximately the
free-fall frame
of the CM
of the Earth.
Are there fundamental inertial frames?
Isaac Newton (1643--1727)
postulated that the average
frame of rest of the
fixed stars defined the
fundamental inertial frame
which he called absolute space.
See the fixed stars in the figure below.
So Newton's idea was wrong, but
the inertial frame of the
fixed stars is a very
good inertial frame for many purposes.
In fact, we do NOT now believe in
absolute space
in Newton's sense.
There is NO single fundamental
inertial frame.
The expanding universe
gives us a different picture.
The two figures below illustrate the
expanding universe
which is a general scaling up of UNBOUND systems.
We believe there are an infinite continuum of
fundamental inertial frames.
Each one attached to a point participating in the mean
expansion of the universe.
These can be called
the comoving frames of the expanding universe.
These reference frames
are actually in free fall,
and so are consistent with our discussion
of inertial frames given above
in subsection Inertial Frames.
The region of space
up to about the scale of
a galaxy cluster
or maybe a galaxy supercluster
can be approximated to some adequacy as a single
local basic
comoving frame.
That single big
comoving frame
can be used for describing the overall motions of
galaxy cluster
or galaxy supercluster.
We can approximately determine the
center of mass (CM)
of galaxy clusters
or galaxy superclusters, and so
identify
their comoving frame
For the
observable universe
as whole, we are beyond the realm of validity of the
concept of
inertial frame
and use general relativity
in a more direct sense.
We can actually identify
the comoving frames
of the expanding universe
which are fundamental
inertial frames.
The barycenters
of most
galaxy clusters
and the reference frame that moves with them
are approximately
comoving frames---this
hypothesis is verified
by the consistency of all modern cosmology.
We can also identify very exactly the local-to-us or nearby-to-us
comoving frames by observations.
Cosmologically remote astronomical objects
(i.e., galaxies
or quasars)
should be unrotating relative to the
local-to-us comoving frames in
modern theory cosmological theory,
and so define for us local unrotating frames.
For the determinations
translational motions
relative to local
comoving frames
for the Sun, the Earth,
and other astronomical objects,
see the figure below.
For very exact, modern studies of motions, one actually does make use of
local comoving frames.
Fortunately, we do NOT usually have to work that hard.
Before general relativity,
the concept of
inertial frame
was a bit mysterious.
An inertial frame
was a reference frame
to which Newtonian physics
was referenced.
But what made
inertial frames
inertial frames
and what made
non-inertial frames
non-inertial frames?
Newton's bucket argument helped
a bit, but it's too intricate to go into here.
The modern defintion depends on the
strong equivalence
principle
of general relativity which
has been verified to a very high degree since 2018:
see the figure below.
Below, we give 2 versions of modern definition of the
the modern inertial frame.
F_g = m(vec g), where m is the body's mass.
The meme comes from
Animal Farm (1945).
See the figure below.
If a reference frame is
inertial enough for your purposes,
it is an inertial frame.
If NOT, it's a non-inertial frame.
There is NO hard line.
Now a force is a physical relationship
between bodies or between a body and
force field
(e.g., the gravitational field
and electromagnetic field)
that causes an
acceleration of a body relative
to all inertial frames.
If you know the forces
acting on a body from known force laws, then
physical law
will predict the acceleration
relative to the inertial frame you are using.
The physical law
in the classical limit is
Newton's 2nd law of motion (AKA F=ma).
If you are NOT
in the classical limit,
you have to use
relativistic mechanics
and/or quantum mechanics.
Newtonian physics
is based on
Newton's 3 laws of motion,
Newton's law of universal gravitation,
and other force laws.
There is whole lot more
Newtonian physics
formalism developed on that basis too.
Actually, almost all physical laws
are referenced to
inertial frames.
But you say we are NOT in free fall
on the surface of the Earth, so how
can we treat the
surface of the Earth as an
inertial frame.
See subsection Center of Mass below for
an explication of center of mass (CM).
For the Earth, it is just the
center of the Earth.
Also see the subsection
Why Do We Need the Concept of Center of Mass?
for why we need the concept of center of mass.
The Earth's gravitational field
is regarded as an internal
gravitational field
of the
CM
free-fall frame
of the Earth.
The Earth's
centrifugal force
is due to Earth's rotation
and it causes an effective reduction to
Earth's gravity.
The centers of mass
for objects of sufficiently high symmetry are the obvious centers of symmetry
as the figure below illustrates.
For objects where
center of mass
CANNOT be found by
by inspection,
one can do a calculation
from the formula
for center of mass displayed in the figure shown
somewhere above.
The center of mass can located
deceptively as shown in the figure below.
Why do we need center of mass
in everyday life?
This inertial frame
is a
free-fall frame orbiting the
center of mass
of the
Milky Way
which defines the
Milky Way
inertial frame.
The most important of the purposes referred to above is just analyzing the motions of the
astro-bodies
that orbit
the Sun.
The moons orbit
their parent planets
in the approximate
inertial frames
(i.e., free-fall frames)
of the
centers of mass of
the planet-moon
systems.
From purely geometrical perspective, motion is all relative.
So you could say either body
in a gravitationally-bound
gavitational two-body system
orbits
the other.
Recall, it is the rotation which holds the
orbiting astronomical objects
from collapsing under gravity.
Where does the NESTED HIERARCHY top out at?
For example, the Earth
in the external
gravitational field
of the Sun
and Moon.
According to general relativity,
Newtonian physics
should apply in this
CM
free-fall frame---as long as
we are in the classical limit: i.e.,
dealing with motions that don't need
general relativity
(needed for strong gravity),
special relativity
(needed for high velocity),
or quantum mechanics
(needed for the
microscopic
particles).
m*vec a_total = vec F_total ,
where m is the mass of a body,
vec a_total is the total
acceleration of the
body relative
to the CM
free-fall frame
and vec F_total is the total force
on the body, except that due to the
gravitational field
right at the CM---its effect
has been canceled out by using the
to the CM
free-fall frame according
general relativity.
m*vec a_total = m*(vec a + vec a_f)
and
vec F_total = vec F + m*vec Δg_e ,
where vec a is the acceleration of the body relative to the
local reference frame we want to use,
vec a_f the acceleration
of the local reference frame,
and Δg_e is the difference in the external
gravitational field
from the external
gravitational field
at the CM.
m*vec a = vec F + m*vec Δg_e - m*vec a_f = vec F + m*(vec Δg_e - vec a_f) ,
where
m*vec Δg_e is called the
tidal force
and -m*vec a_f the inertial force---but
it's NOT a real force---it just acts like one.
But, of course, we now know that the fixed stars
are all very nearby in the Milky Way
and do NOT represent the observable universe.
Note the
expanding universe
is the observable universe
and probably a lot more, but NOT necessarily the
whole universe whatever that is.
Maybe the whole universe is
the multiverse.
Yours truly has invented the expression
pan-universe for the
whole universe, but
yours truly doesn't think it will catch on.
Note that systems like you, me,
moons,
planets,
stars,
galaxies, and
probably most galaxy clusters
are NOT expanding.
But the space
between them is.
In general relativity,
space is a sort of stuff and
it can literally grow.
Nowadays from general relativity,
we do NOT believe in a single
fundamental inertial frame
(i.e., a fundamental fundamental
inertial frame).