AA-U1-W3-ML1 Physics of Orbits 1 - Energy
Summary
TLDRThis educational script delves into the physics of orbital motion, contrasting the 'forces approach' using Newton's laws with the 'energy approach' focusing on conservation of energy. It explains how the total energy of a system, combining kinetic and potential energies, can predict the behavior of astronomical objects in orbit. The script introduces concepts like angular momentum and effective potential energy, illustrating how they dictate the shape of orbits and the stability of circular paths. It also touches on the challenges of sending probes close to massive objects due to energy barriers and the importance of angular momentum in preventing objects from spiraling into the central mass.
Takeaways
- 🔍 There are two primary approaches to solving physics problems: the forces approach, which uses Newton's laws, and the energy approach, which applies the conservation of energy.
- 📚 The forces approach involves solving differential equations derived from Newton's second law, often starting with simple scenarios like constant velocity or acceleration.
- 🔄 The energy approach sums the kinetic and potential energies of a system and uses the conservation of energy to solve for system states, often involving integral equations.
- 🌌 In orbital mechanics, the energy method helps analyze the motion of astronomical objects by considering their kinetic and potential energies.
- 📉 The effective potential energy, which includes both potential and a component of kinetic energy, is crucial for understanding orbits and is plotted against radius to visualize orbital behavior.
- ⚫ The effective potential energy curve shows an infinite barrier at the center, preventing objects from spiraling into the central mass, illustrating the conservation of angular momentum.
- 🔵 At the minimum of the effective potential energy curve, an object will maintain a constant radius (Rc), indicative of a stable circular orbit.
- 🔴 The location of the minimum (Rc) in the effective potential energy curve depends on the angular momentum of the orbiting object and the mass of the central body.
- 🔄 Perturbations from the minimum radius result in elliptical orbits, where the radial coordinate executes simple harmonic motion, approximating an elliptical path.
- 📈 The energy method provides a clear picture of how objects in orbit respond to changes in energy and angular momentum, predicting stable orbits and the effects of perturbations.
Q & A
What are the two general approaches to solving problems in physics?
-The two general approaches to solving problems in physics are the forces approach and the energy approach. The forces approach involves determining all the forces acting on a system and applying the equations of motion, typically Newton's laws. The energy approach involves writing down the total energy of the system as a sum of kinetic and potential energy and applying the law of conservation of energy.
How does the forces approach typically use Newton's second law of motion?
-The forces approach uses Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object times its acceleration. This approach often results in differential equations that need to be solved to find the position, velocity, and acceleration of an object over time.
What is the significance of angular momentum in orbital motion?
-Angular momentum is a key characteristic of an orbit and plays a crucial role in determining the shape and stability of an orbit. It prevents an object from spiraling into the central body and acts as an energy barrier. The conservation of angular momentum dictates that an object with any tangential velocity component cannot fall into the central object.
How is the energy per unit mass of a test particle orbiting a central mass derived?
-The energy per unit mass (epsilon) of a test particle orbiting a central mass is derived by considering the kinetic energy of the particle and the gravitational potential energy. It is expressed as epsilon = 1/2 * V_R^2 + L^2 / (2 * m * R^2) - G * M / R, where V_R is the radial velocity, L is the angular momentum, m is the mass of the test particle, R is the radial distance, and G * M is the gravitational potential energy.
What does the effective potential energy represent in the context of orbital motion?
-The effective potential energy in the context of orbital motion represents a combination of the actual potential energy due to gravity and the potential energy associated with the angular momentum of the orbiting object. It is useful for visualizing and understanding the behavior of orbits, especially in terms of stability and the tendency of objects to move towards energy minima.
Why is it difficult for a probe to get close to the Sun?
-It is difficult for a probe to get close to the Sun because it must overcome the energy barrier created by the conservation of angular momentum. The probe needs to have sufficient energy to either climb up the centrifugal energy barrier or burn away its angular momentum to spiral in towards the Sun, which requires a colossal amount of fuel.
What is the significance of the minimum in the effective potential energy curve?
-The minimum in the effective potential energy curve corresponds to the most stable orbit, where the total energy of the system is minimized. This minimum represents the radius at which an object will naturally tend to stay if placed there, and it is the radius of a circular orbit for a given amount of angular momentum.
How does the concept of simple harmonic motion relate to orbital motion?
-In orbital motion, if an object is placed close to the minimum of the effective potential energy curve, it will execute simple harmonic motion in the radial direction. This means that the radial distance from the central object will oscillate back and forth in a sinusoidal pattern, resulting in an elliptical orbit.
What is perturbation theory and how does it relate to solving differential equations in orbital motion?
-Perturbation theory is a mathematical method used to approximate the solution to a problem by starting with a known solution to a simpler problem and then making small adjustments or 'perturbations' to account for the complexities of the actual problem. In the context of orbital motion, it can be used to solve differential equations by considering small deviations from a known, simpler orbit, such as a circular orbit.
What is the relationship between the conservation of energy and Newton's second law in the context of orbital motion?
-In the context of orbital motion, the conservation of energy and Newton's second law are closely related. The conservation of energy principle states that the total energy of a system remains constant, which can be expressed in terms of the effective potential and kinetic energy. Differentiating the total energy with respect to a variable (like radius) and setting it to zero, as required by the conservation of energy, leads to an equation that is essentially Newton's second law, showing the force acting on the object.
Outlines
🪐 Introduction to Orbital Motion and Problem-Solving Approaches
In this paragraph, the speaker introduces two common methods to solve problems in physics: the forces approach and the energy approach. The forces approach uses Newton’s laws, particularly focusing on Newton’s second law, to calculate position, velocity, and acceleration over time. On the other hand, the energy approach leverages the conservation of energy principle, where the total energy at the beginning equals the total energy at the end. This method often involves solving integral equations, particularly useful in complex situations. The speaker also explains how these methods apply to celestial orbits.
🔺 Kinetic and Potential Energy in Orbital Mechanics
This paragraph explores how kinetic and potential energies interact in orbital systems. The kinetic energy of an object in orbit is described as 1/2 MV^2, where V is the magnitude of velocity, which can be broken into radial and angular components. Potential energy is expressed using gravitational constants. Additionally, the paragraph introduces angular momentum as a crucial factor in orbit dynamics. Angular momentum is calculated using mass, radial distance, and angular velocity, leading to a key equation for total energy, combining kinetic and potential components.
📉 Effective Potential Energy and Orbit Shapes
Here, the speaker discusses how effective potential energy plays a role in determining orbital motion. The effective potential energy combines radial kinetic energy and gravitational potential energy. The speaker sketches a graph of potential energy versus radius, explaining its behavior as radius approaches zero and infinity. This graph demonstrates a minimum energy point, which helps explain stable orbits, including elliptical and hyperbolic trajectories, and the energy barriers that prevent objects from spiraling into a central mass, such as the Sun.
⚖ Deriving the Minimum of Effective Potential Energy
In this section, the speaker explains how to find the minimum of effective potential energy by differentiating the equation and solving for the radial distance (Rc) at which energy is minimized. The derived formula shows that this radius depends on the angular momentum and mass of the central object. The paragraph highlights that when an object is at this minimum energy point, it remains in a stable circular orbit. This result also explains why a planet or spacecraft with a specific angular momentum can only have one corresponding circular orbit.
🔄 Oscillations Around Circular Orbits
This paragraph delves into what happens when an object deviates slightly from the minimum energy point. If an orbit starts slightly outside or inside the stable radius, the object will execute simple harmonic motion as it oscillates between smaller and larger radial distances. The speaker describes this motion as an elliptical orbit with sinusoidal perturbations. The explanation provides insight into how elliptical orbits form from deviations in radial velocity and the overall energy balance in the system.
🔍 Energy-Based Approach to Orbital Motion
In the final paragraph, the speaker sums up the utility of the energy approach in analyzing orbits. While this method offers clear insights into key aspects like potential energy barriers and orbital stability, the speaker also mentions the importance of solving differential equations. They introduce the concept of perturbation theory, hinting that this technique will be explored further in the next lecture to solve more complex orbital dynamics.
Mindmap
Keywords
💡Orbits
💡Forces Approach
💡Energy Approach
💡Newton's Laws
💡Gravitational Potential Energy
💡Angular Momentum
💡Conservation of Energy
💡Effective Potential Energy
💡Perturbation Theory
💡Circular Orbit
💡Elliptical Orbit
Highlights
Two primary approaches in physics: the forces approach and the energy approach.
Forces approach involves applying Newton's laws to solve for position, velocity, and acceleration.
Energy approach involves writing the total energy of the system and applying the law of conservation of energy.
Gravitational potential energy for spherical objects is calculated by integrating over the mass of small constituents.
Orbital motion can be analyzed using energy approach with central mass and orbiting object.
Velocity in orbital motion has both tangent and radial components, denoted by V_Phi and V_R.
Angular momentum (L) is a key characteristic of an orbit, calculated as mass times angular velocity.
Total energy of an orbiting particle is the sum of kinetic and potential energy.
Energy per unit mass is a useful quantity for describing the orbit of a test particle.
Effective potential energy is a concept that combines certain terms of kinetic and potential energy for analysis.
The effective potential energy has a minimum, which corresponds to the stable circular orbit.
The location of the minimum (RC) depends on the angular momentum and the mass of the central object.
Objects in orbit tend to move towards the minimum of their total energy.
If a particle's orbit starts with a radius different from RC, it will execute simple harmonic motion around the minimum.
The radial coordinate in an elliptical orbit can be described by a sinusoidal function of time.
The energy method provides a straightforward way to derive the basics of orbital motion.
Perturbation theory will be introduced to solve the differential equation of orbital motion in the next lecture.
Transcripts
I'd like to talk a little bit more about
orbits and how you go about solving the
orbits the orbital motion of
astronomical objects and I want to make
a few general comments first in pretty
much any area of physics there are
generally two ways to approach a problem
okay there is what you could call the
forces approach or the energy approach
and in the forces approach simply put
you figure out all the forces acting on
a system and you apply the equations of
motion
usually newton's laws or some
generalization of them in order to solve
for the position velocity and
acceleration as a function of time okay
so the forces approach typically uses or
is basically based off of Newton's
second law of motion some of the forces
is equal to mass times acceleration
which is the second derivative of
position okay and so generally this
results in differential equations that
you need to solve now of course when we
start learning about this approach we
generally choose very simple situations
such as constant velocity or constant
acceleration in which case the
differential equations usually simplify
and their simplest just become algebraic
equations but this is the general
approach you order the tops now you can
go through an equivalent but different
procedure that is sometimes easier to
solve and that is you deal with the
energy of the system so you write down
the total energy of the system which is
a sum of the kinetic energy and
potential energy system and you apply
the law of conservation of energy
so the energy at the start of some
process is equal to the energy at the
ends of the process that gives you an
equation which you then solve okay so in
these cases the laws of physics that
we're using in this case is essentially
Newton's second law of motion and in
this cases a conservation of energy
now energy the energy of a system is the
sum total of the energy of its
constituents okay for example we will
see in a few weeks that the
gravitational potential energy for
example of a spherical object like the
earth or the Sun is simply the
gravitational potential energy of one
small constituent of that sphere DM DM
is just a small constituent the mass of
a small constituent of the sphere
divided by R so this is the just the
Newtonian formula for gravitational
potential energy and then you have to
integrate this you have to add up every
piece of mass in the earth or the Sun or
whatever system it is you are looking at
so because to find energy involves
summing or integrating then this energy
approach in general involves solving
integral equations and that sometimes
can be easier than solving differential
equations and sometimes harder and you
tend to just pick which approach to use
based on the situation and based on your
experience
so with those initial remarks let's turn
to orbits I'm going to start off working
in the energy approach and let's say we
have a central mass M and we have our
object orbiting the mass and it's
orbiting it with some velocity V now in
general we're going to deal with we're
not going to assume it's going to be a
circular orbit it can be in general and
elliptical orbit there also
one can also have hyperbolic and
parabolic orbits and and that means that
in general the velocity is not tangent
to the orbital path like it is in
circular motion so in other words whilst
there is a component which I'm going to
call V sub Phi that is in the direction
tangent to the orbital path there will
also be a component that is radially in
or radially outwards
okay now I call this Phi because it's
the most convenient coordinates to use
are the radial coordinate ah and an
angular coordinate Phi which just
measures the angle relative to some
arbitrary point on the elliptical path
which we can choose to be one of them
whatever we want okay now the we can
write down the energies the energy of
this particle its kinetic energy is
simply one-half MV squared where V is
the magnitude of the velocity V and by
Pythagoras theorem this is just the sum
of V Phi squared and V R squared and
then the potential energy is equal to
minus g-m Big M little m that's called
the mass of this little m / ah okay now
it's also useful - and not just useful
it is very important to know the angular
momentum of the orbit the angular
momentum is a key characteristic of the
orbit and that which we'll call capital
L is equal to the mass times the angular
speed of the orbital angular velocity of
the orbit which is so we get M R V sub
Phi and we can rearrange this we can
solve for V sub Phi is equal to L
divided by M R and therefore V Phi
squared is equal to L square divided by
M Squared R squared
okay now we do this so that when we
write our total energy let's write down
the total energy so we've got the
kinetic energy 1/2 M let's write the
radial component first
plus 1/2 MV Phi squared minus G Big M
little m / R potential energy and this
is equal to 1/2 MV R squared + now we
can substitute V Phi squared here for l
squared over m squared R squared and
that gives us for this term l squared
divided by 2m R squared and then minus
GMM over R and furthermore it is useful
to simplify this a little further by
defining the energy per unit mass of our
test particle of our particle that is
orbiting the the central mass so this if
we divide by little m we get 1/2 V R
squared plus l squared divided by 2 m
squared R squared minus G Big M over R
and then finally we can define the
angular momentum per unit mass as the
angular momentum divided by the mass so
this term here big L square divided by
with lems squared is just little l
squared and we're going to let me call
this quantity Epsilon
let's go just to find epsilon to be a /
M then we end up with the equation I'm
gonna write this that's off of this next
page epsilon is equal to one-half V R
squared plus little L square divided by
2 R squared minus G M over R okay so
this is the energy per unit mass of the
test particle for example if the
particle has a mass of one kilogram
exactly then this is just its energy so
this is completely general for a
completely general expression for the
the energy per mass of a test particle
orbiting
mass no matter what the shape of the
orbit okay this is valid for elliptical
orbits circular orbits like a Pollock
orbits and so on
now it's useful to actually break this
up into two parts or to think of it as
two parts this part here is let's call
it K tilde and this is the radial
kinetic energy and then I'm going to
call these two parts here the effective
potential energy V tilde effective and
it'll become more apparent why I'm
lumping this term in with as part of an
effective potential energy even though
it originated as part of the kinetic
energy it actually becomes useful to
view it in that way and it is maybe a
little easier to see why it's useful if
we actually plot the effective potential
energy versus the radius R so let's
sketch a plot of V tilde effective
versus R okay so to sketch a graph of
this just to figure out what its shape
is the the standard technique is firstly
look at how it behaves in the limits of
are tending to zero and are tending to
infinity well as R tends to zero then
the term here with the highest power of
R in the denominator is going to win
okay the smaller art becomes then the
bigger
for example this becomes compared to
this yeah so as R tends to 0 this term
wins and so the effective behaves as 1
over R squared and 1 over R squared
shoots off to infinity as R tends to 0
now when R becomes very big then it's
the term with the smallest power of R in
the denominator that will win because
this will vanish quicker than this as R
becomes really big so in the limit as R
tending tends to infinity this term wins
and we can neglect this term so this is
minus one over R and minus one over R as
R tends to infinity goes to zero from
below the R axis like that okay so we
know how this curve behaves in the two
limits and we know that of course they
must smoothly join together somewhere in
the middle and the only way to do that
is for this to come down here and for
there to be a minimum at some point and
for it to turn around like that so this
potential energy looks like this and the
first thing we can note is that it
shoots off to infinity here okay and no
matter how much kinetic energy you have
associated with the radial motion like
so in other words no matter how fast you
manage to shoot your op your test mass
in this way no matter how high this term
is here let's mark this will have some
value that we can also plot on the same
axis since they're both energies it will
the effect this effective potential
energy will always increase above this
contribution to the kinetic energy so
long as you are close enough into the
object and by conservation of energy
that means that you can if you start off
with the total amount of kinetic energy
it is going to always turn into back
into potential energy you're going to
lose that kinetic energy to potential
energy at some point before you hit R
equal to zero in other words this is an
energy barrier that prevents any object
getting to R equal to zero in other
words so long as you have any component
of velocity tangent to the radial
direction or sorry perpendicular to the
radial direction tangent to the orbital
motion then it is impossible to ever
fall and spiral in and hit the central
object you just you have to give it an
infinite radial velocity to do that
right this is just what we know as the
center of
Google energy barrier and this is a
manifestation of that you know what is
sometimes known as a fictitious force
which prevents you from losing it's
essentially just a manifestation of the
conservation of angular momentum the
only way to break through this barrier
is if you lose all of your angular
momentum and then this term here which
is the term responsible for it going to
infinity vanishes so the message here is
that any slight velocity component that
is not directed radially will always
prevent any object spiraling in to the
central object you're orbiting around it
is in part this is really really really
difficult to send probes close to the
Sun for this exact reason it costs a
colossal amount of fuel to get a probe a
space probe anywhere close to the Sun
because you are having to essentially
give it sufficient energy to climb up
the centrifugal energy barrier or at
least burn away your angular momentum so
that's one important aspect of orbits
that comes out when we break up the
energy in this way so this essentially
if it acts as an effective potential
barrier the second thing we notice that
it has a minimum here which I'll call RC
and the another overriding principle of
physics is that things systems tend to
want to move towards the minimum in
their total energy they want to minimize
their energy and if they're already
there they're not coming out of there
and so if you are at this location here
RC then you're going to tend to want to
stay there and let's see how this comes
out in the mathematics
so that's first find the location of
this minimum to do that we simply take
the derivative of this and set it equal
to zero so we have let me just write
down v effective on its own it's equal
to little l squared over 2 r squared
minus g-m over r differentiate it with
respect to r and you'll get minus l
squared over r cubed plus G M over R
squared and we want to then set this
equal to 0 this gives minus l squared
over R cubed plus GM over R squared is
equal to 0 multiplied by R cubed and you
get minus l squared plus G M R is equal
to 0 now let's say let's we know that
it's 0 when when I was equal to R see as
I've called it so that's just like that
subscript there and then we can
rearrange this to find r c is equal to l
squared over G M ok so this the radius
here the location where the energy is a
minimum depends on the angular momentum
and the mass of the central object the
angular momentum of the test particle
and the mass of the central object okay
so let's just check what happens when
you put a particle at this distance
well to do that let's what we did here
is we differentiated the effective this
effective potential energy let's let's
differentiate the whole thing the whole
energy with respect to R and to do that
we need to know how to differentiate V
sub R with respect to R well V sub R is
just the R by DT right it is the radial
velocity the rate of change of the
radial coordinate with respect to time
and the kinetic energy is 1/2 the clinic
the kinetic energy per unit mass is just
1/2 V R squared which is 1/2 dr by dt
squared and therefore if we
differentiate this with respect to R
then we can do this by using the chain
rule we can write D K tilde by dr as
decay tilde by DT times DT by dr okay
and decay tilde by dt if we
differentiate this with respect to time
we are going to get first we
differentiate the contents of these
parentheses which gives two times dr by
dt and that 2 cancels and then we
multiply that by the differential of dr
by dt with respect to t this is d 2 R by
DT squared so this is d K tilde by DT
and then we multiplied by DT by dr but
dt by dr is the inverse of dr by dt and
so we just get D 2 R by DT squared
therefore
if we differentiate the total energy
with respect to power then this is DK
tilted by dr plus DV tilde effective by
dr and this is equal to D to R by DT
squared plus DV tilde effective by dr
what is the differential of the total
energy with respect to anything it's
zero that's conservation of energy total
energy doesn't change like so it's
variation with respect to anything is
always zero so this is just conservation
of energy and we end up with D to R by
DT squared is equal to minus D V tilde
effective by dr this is just Newton's
second law so this actually illustrates
more generally that Newton's second law
can be derived directly from the law of
conservation of energy so conservation
of energy and Newton's second law are
not really different things this is our
effective force which contains of course
the real force of gravity if we
differentiate this with respect to R you
get Newton's universal law of gravity
but then it also contains this and this
is the what we sometimes think of as a
fictitious force so going back to our
original point we were interested in
this point where the derivative of the
effective potential respect R is zero
this point here well if it's zero then D
- R by DT squared is equal to zero there
is no acceleration there's no
acceleration in words for outwards okay
so if if you start off with R is equal
to R see there
and solve this equation you'll find that
are at all future times is equal to rc
in other words R is a constant and R is
a constant is a circular orbit so this
confirms what we probably suspected that
this minimum here corresponds to the
circular orbit what this tells us the
fact that it's related to the angular
momentum and the mass of the central
object is that if you give a planet or a
spacecraft or anything a certain amount
of angular momentum right then there is
only one possible radius only one
possible distance it can orbit at in a
circular orbit there's only one circular
orbit for a given amount of angular
momentum okay now you could ask well
what happens then if I start off with an
orbit that is slightly different from RC
so let's suppose I start out here okay
what happens then well what's gonna
happen is it's going to try to find its
way back in towards the center here okay
it's gonna find its what you're gonna
try and find its way back in to the
minimum and when it gets there it's
gonna overshoot because it's acquired a
certain radial velocity so it's going to
come back out the other side and then
it's gonna as it climbs up this
potential well it's going to lose some
radial kinetic energy until its radial
kinetic energy is zero and then it comes
back down here and then up again so
execute simple harmonic motion so long
as you put it fairly close to the
minimum I close enough that you can
approximate this curve here as a
quadratic right that's quadratic in are
how do we see that well again we can go
back to our equation here if this is if
the tilde effective is quadratic in R
then its differential is linear
and you simply get D to R by DT squared
is equal to minus some constant times R
right so this is just the equation of
simple harmonic motion for the radial
coordinate okay so in other words the
radial coordinate will execute simple
harmonic motion and we'll derive this
again in a second so in other words R
can in general be written as some
constant times sine of KT square root of
KT where K is the frequency that
oscillates back and forth now what this
is just the radial coordinate what does
that look like for an orbit well here's
the orbit let's try and draw the sake of
orbit first of all so this is the
circular orbit this orbit is R is equal
to RC a constant now if it is executing
this motion here then what we're saying
is we're starting slightly further away
than the circular orbit and then as time
goes on and it moves around the radius
is going to come in and it's going to
enter and it's going to cross over RC
and it's going to become smaller than RC
and then it's going to move back out
again so this is just I should put a
plus RC here this is just orbiting with
a sinusoidal perturbation on the
circular orbit so we have an elliptical
orbit
so using the energy method is a nice
fairly straightforward way to derive
some of the basics of orbital motion and
to see how that looks like what that
looks like in the energy picture but
it's also useful to try and solve the
differential equation and in doing so I
want to introduce you to a little bit of
perturbation theory which I will do in
the next lecture
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