AA-U1-W3-ML1 Physics of Orbits 1 - Energy

Summary

This 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

00:00

🪐 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.

05:00

🔺 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.

10:06

📉 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.

15:08

⚖ 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.

20:08

🔄 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.

25:08

🔍 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.

Keywords

💡Orbits

Orbits refer to the paths that astronomical objects, such as planets and satellites, follow around celestial bodies due to the influence of gravity. In the script, orbits are central to understanding the motion of objects in space, with a focus on how they are influenced by forces and energy. The discussion includes different types of orbits like circular, elliptical, hyperbolic, and parabolic.

💡Forces Approach

The forces approach is a method in physics for solving problems by determining all the forces acting on a system and applying the equations of motion, typically Newton's laws. In the context of the video, this approach is contrasted with the energy approach as a way to analyze orbital motion, emphasizing the calculation of position, velocity, and acceleration over time.

💡Energy Approach

The energy approach involves calculating the total energy of a system, which is the sum of its kinetic and potential energies, and applying the law of conservation of energy. This method is highlighted in the script as an alternative to the forces approach, particularly useful for solving problems involving complex motion, like orbital paths, by simplifying the equations through energy conservation.

💡Newton's Laws

Newton's laws of motion are the fundamental principles that describe the relationship between the motion of an object and the forces acting upon it. In the script, Newton's second law, which states that force equals mass times acceleration, is specifically mentioned as the basis for the forces approach in analyzing orbital motion.

💡Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. The script discusses how this energy is calculated for spherical objects like the Earth or the Sun by integrating the potential energy of their individual mass constituents, which is crucial for understanding the energy dynamics in orbital motion.

💡Angular Momentum

Angular momentum is a measure of the rotational motion of an object and is a key characteristic in orbital mechanics. The script emphasizes its importance in determining the shape and stability of orbits, as well as the fact that any object in orbit possesses angular momentum that prevents it from spiraling into the central body.

💡Conservation of Energy

The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In the script, this principle is applied to analyze the total energy of a system, helping to derive equations that describe the behavior of objects in orbit, such as the conditions for stable circular orbits.

💡Effective Potential Energy

Effective potential energy is a concept used in the script to combine the potential energy and a component of the kinetic energy into a single function that helps visualize and understand the stability and behavior of orbits. It is particularly useful for explaining why objects in orbit do not spiral into the central body and for predicting the shape of orbits.

💡Perturbation Theory

Perturbation theory is a mathematical method used to approximate the solution of a complex system by making small changes to a simpler, known system. The script mentions that this theory will be introduced to solve differential equations related to orbital motion, particularly useful for analyzing deviations from idealized orbits.

💡Circular Orbit

A circular orbit is a specific type of orbit where an object moves in a perfect circle around a central body. The script discusses how the energy approach can be used to determine the conditions for a stable circular orbit, which is dependent on the object's angular momentum and the mass of the central body.

💡Elliptical Orbit

An elliptical orbit is another common type of orbit where the path of the object is an ellipse, with the central body at one of the two foci. The script explains how elliptical orbits can be derived from the energy method, showing that slight deviations in energy or angular momentum from a circular orbit can result in an elliptical path.

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