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Chapter 11

- Gravity, Planetary Orbits, and
- the Hydrogen Atom

Newtons Law of Universal Gravitation

- Every particle in the Universe attracts every

other particle with a force that is directly

proportional to the product of their masses and

inversely proportional to the square of the

distance between them - G is the universal gravitational constant and

equals 6.673 x 10-11 N?m2 / kg2

Law of Gravitation, cont

- This is an example of an inverse square law
- The magnitude of the force varies as the inverse

square of the separation of the particles - The law can also be expressed in vector form

Notation

- is the force exerted by particle 1 on

particle 2 - The negative sign in the vector form of the

equation indicates that particle 2 is attracted

toward particle 1 - is the force exerted by particle 2 on

particle 1

More About Forces

- The forces form a Newtons Third Law

action-reaction pair - Gravitation is a field force that always exists

between two particles, regardless of the medium

between them - The force decreases rapidly as distance increases
- A consequence of the inverse square law

G vs. g

- Always distinguish between G and g
- G is the universal gravitational constant
- It is the same everywhere
- g is the acceleration due to gravity
- g 9.80 m/s2 at the surface of the Earth
- g will vary by location

Gravitational Force Due to a Distribution of Mass

- The gravitational force exerted by a

finite-sized, spherically symmetric mass

distribution on a particle outside the

distribution is the same as if the entire mass of

the distribution were concentrated at the center - For the Earth, this means

Measuring G

- G was first measured by Henry Cavendish in 1798
- The apparatus shown here allowed the attractive

force between two spheres to cause the rod to

rotate - The mirror amplifies the motion
- It was repeated for various masses

Gravitational Field

- Use the mental representation of a field
- A source mass creates a gravitational field

throughout the space around it - A test mass located in the field experiences a

gravitational force - The gravitational field is defined as

Gravitational Field of the Earth

- Consider an object of mass m near the earths

surface - The gravitational field at some point has the

value of the free fall acceleration - At the surface of the earth, r RE and g 9.80

m/s2

Representations of the Gravitational Field

- The gravitational field vectors in the vicinity

of a uniform spherical mass - fig. a the vectors vary in magnitude and

direction - The gravitational field vectors in a small region

near the earths surface - fig. b the vectors are uniform

Structural Models

- In a structural model, we propose theoretical

structures in an attempt to understand the

behavior of a system with which we cannot

interact directly - The system may be either much larger or much

smaller than our macroscopic world - One early structural model was the Earths place

in the Universe - The geocentric model and the heliocentric models

are both structural models

Features of a Structural Model

- A description of the physical components of the

system - A description of where the components are located

relative to one another and how they interact - A description of the time evolution of the system
- A description of the agreement between

predictions of the model and actual observations - Possibly predictions of new effects, as well

Keplers Laws, Introduction

- Johannes Kepler was a German astronomer
- He was Tycho Brahes assistant
- Brahe was the last of the naked eye astronomers
- Kepler analyzed Brahes data and formulated three

laws of planetary motion

Keplers Laws

- Keplers First Law
- Each planet in the Solar System moves in an

elliptical orbit with the Sun at one focus - Keplers Second Law
- The radius vector drawn from the Sun to a planet

sweeps out equal areas in equal time intervals - Keplers Third Law
- The square of the orbital period of any planet is

proportional to the cube of the semimajor axis of

the elliptical orbit

Notes About Ellipses

- F1 and F2 are each a focus of the ellipse
- They are located a distance c from the center
- The longest distance through the center is the

major axis - a is the semimajor axis

Notes About Ellipses, cont

- The shortest distance through the center is the

minor axis - b is the semiminor axis
- The eccentricity of the ellipse is defined as e

c /a - For a circle, e 0
- The range of values of the eccentricity for

ellipses is 0 lt e lt 1

Notes About Ellipses, Planet Orbits

- The Sun is at one focus
- Nothing is located at the other focus
- Aphelion is the point farthest away from the Sun
- The distance for aphelion is a c
- For an orbit around the Earth, this point is

called the apogee - Perihelion is the point nearest the Sun
- The distance for perihelion is a c
- For an orbit around the Earth, this point is

called the perigee

Keplers First Law

- A circular orbit is a special case of the general

elliptical orbits - Is a direct result of the inverse square nature

of the gravitational force - Elliptical (and circular) orbits are allowed for

bound objects - A bound object repeatedly orbits the center
- An unbound object would pass by and not return
- These objects could have paths that are parabolas

- and hyperbolas

Orbit Examples

- Pluto has the highest eccentricity of any planet

(a) - ePluto 0.25
- Halleys comet has an orbit with high

eccentricity (b) - eHalleys comet 0.97

Keplers Second Law

- Is a consequence of conservation of angular

momentum - The force produces no torque, so angular momentum

is conserved

Keplers Second Law, cont.

- Geometrically, in a time dt, the radius vector r

sweeps out the area dA, which is half the area of

the parallelogram - Its displacement is given by

Keplers Second Law, final

- Mathematically, we can say
- The radius vector from the Sun to any planet

sweeps out equal areas in equal times - The law applies to any central force, whether

inverse-square or not

Keplers Third Law

- Can be predicted from the inverse square law
- Start by assuming a circular orbit
- The gravitational force supplies a centripetal

force - Ks is a constant

Keplers Third Law, cont

- This can be extended to an elliptical orbit
- Replace r with a
- Remember a is the semimajor axis
- Ks is independent of the mass of the planet, and

so is valid for any planet

Keplers Third Law, final

- If an object is orbiting another object, the

value of K will depend on the object being

orbited - For example, for the Moon around the Earth, KSun

is replaced with KEarth

Energy in Satellite Motion

- Consider an object of mass m moving with a speed

v in the vicinity of a massive object M - M gtgt m
- We can assume M is at rest
- The total energy of the two object system is E

K Ug

Energy, cont.

- Since Ug goes to zero as r goes to infinity, the

total energy becomes

Energy, Circular Orbits

- For a bound system, E lt 0
- Total energy becomes
- This shows the total energy must be negative for

circular orbits - This also shows the kinetic energy of an object

in a circular orbit is one-half the magnitude of

the potential energy of the system

Energy, Elliptical Orbits

- The total mechanical energy is also negative in

the case of elliptical orbits - The total energy is
- r is replaced with a, the semimajor axis

Escape Speed from Earth

- An object of mass m is projected upward from the

Earths surface with an initial speed, vi - Use energy considerations to find the minimum

value of the initial speed needed to allow the

object to move infinitely far away from the Earth

Escape Speed From Earth, cont

- This minimum speed is called the escape speed
- Note, vesc is independent of the mass of the

object - The result is independent of the direction of the

velocity and ignores air resistance

Escape Speed, General

- The Earths result can be extended to any planet
- The table at right gives some escape speeds from

various objects

Escape Speed, Implications

- This explains why some planets have atmospheres

and others do not - Lighter molecules have higher average speeds and

are more likely to reach escape speeds - This also explains the composition of the

atmosphere

Black Holes

- A black hole is the remains of a star that has

collapsed under its own gravitational force - The escape speed for a black hole is very large

due to the concentration of a large mass into a

sphere of very small radius - If the escape speed exceeds the speed of light,

radiation cannot escape and it appears black

Black Holes, cont

- The critical radius at which the escape speed

equals c is called the Schwarzschild radius, RS - The imaginary surface of a sphere with this

radius is called the event horizon - This is the limit of how close you can approach

the black hole and still escape

Black Holes and Accretion Disks

- Although light from a black hole cannot escape,

light from events taking place near the black

hole should be visible - If a binary star system has a black hole and a

normal star, the material from the normal star

can be pulled into the black hole

Black Holes and Accretion Disks, cont

- This material forms an accretion disk around the

black hole - Friction among the particles in the disk

transforms mechanical energy into internal energy

Black Holes and Accretion Disks, final

- The orbital height of the material above the

event horizon decreases and the temperature rises - The high-temperature material emits radiation,

extending well into the x-ray region - These x-rays are characteristics of black holes

Black Holes at Centers of Galaxies

- There is evidence that supermassive black holes

exist at the centers of galaxies - Theory predicts jets of materials should be

evident along the rotational axis of the black

hole

- An HST image of the galaxy M87. The jet of

material in the right frame is thought to be

evidence of a supermassive black hole at the

galaxys center.

Gravity Waves

- Gravity waves are ripples in space-time caused by

changes in a gravitational system - The ripples may be caused by a black hole forming

from a collapsing star or other black holes - The Laser Interferometer Gravitational Wave

Observatory (LIGO) is being built to try to

detect gravitational waves

Importance of the Hydrogen Atom

- A structural model can also be used to describe a

very small-scale system, the atom - The hydrogen atom is the only atomic system that

can be solved exactly - Much of what was learned about the hydrogen atom,

with its single electron, can be extended to such

single-electron ions as He and Li2

Light From an Atom

- The electromagnetic waves emitted from the atom

can be used to investigate its structure and

properties - Our eyes are sensitive to visible light
- We can use the simplification model of a wave to

describe these emissions

Wave Characteristics

- The wavelength, l, is the distance between two

consecutive crests - A crest is where a maximum displacement occurs
- The frequency, ƒ, is the number of waves in a

second - The speed of the wave is c ƒ l

Atomic Spectra

- A discrete line spectrum is observed when a

low-pressure gas is subjected to an electric

discharge - Observation and analysis of these spectral lines

is called emission spectroscopy - The simplest line spectrum is that for atomic

hydrogen

Uniqueness of Atomic Spectra

- Other atoms exhibit completely different line

spectra - Because no two elements have the same line

spectrum, the phenomena represents a practical

and sensitive technique for identifying the

elements present in unknown samples

Emission Spectra Examples

Absorption Spectroscopy

- An absorption spectrum is obtained by passing

white light from a continuous source through a

gas or a dilute solution of the element being

analyzed - The absorption spectrum consists of a series of

dark lines superimposed on the continuous

spectrum of the light source

Absorption Spectrum, Example

- A practical example is the continuous spectrum

emitted by the sun - The radiation must pass through the cooler gases

of the solar atmosphere and through the Earths

atmosphere

Balmer Series

- In 1885, Johann Balmer found an empirical

equation that correctly predicted the four

visible emission lines of hydrogen - H? is red, ? 656.3 nm
- H? is green, ? 486.1 nm
- H? is blue, ? 434.1 nm
- H? is violet, ? 410.2 nm

Emission Spectrum of Hydrogen Equation

- The wavelengths of hydrogens spectral lines can

be found from - RH is the Rydberg constant
- RH 1.097 373 2 x 107 m-1
- n is an integer, n 3, 4, 5,
- The spectral lines correspond to different values

of n

Niels Bohr

- 1885 1962
- An active participant in the early development of

quantum mechanics - Headed the Institute for Advanced Studies in

Copenhagen - Awarded the 1922 Nobel Prize in physics
- For structure of atoms and the radiation

emanating from them

The Bohr Theory of Hydrogen

- In 1913 Bohr provided an explanation of atomic

spectra that includes some features of the

currently accepted theory - His model includes both classical and

non-classical ideas - He applied Plancks ideas of quantized energy

levels to orbiting electrons

Bohrs Assumptions for Hydrogen, 1

- The electron moves in circular orbits around the

proton under the electric force of attraction - The force produces the centripetal acceleration
- Similar to the structural model of the Solar

System

Bohrs Assumptions, 2

- Only certain electron orbits are stable and these

are the only orbits in which the electron is

found - These are the orbits in which the atom does not

emit energy in the form of electromagnetic

radiation - Therefore, the energy of the atom remains

constant and classical mechanics can be used to

describe the electrons motion - This representation claims the centripetally

accelerated electron does not emit energy and

eventually spirals into the nucleus

Bohrs Assumptions, 3

- Radiation is emitted by the atom when the

electron makes a transition from a more energetic

initial state to a lower-energy orbit - The transition cannot be treated classically
- The frequency emitted in the transition is

related to the change in the atoms energy - The frequency is independent of the frequency of

the electrons orbital motion - The frequency of the emitted radiation is given

by - Ei Ef hƒ
- h is Plancks constant and equals 6.63 x 10-34 Js

Bohrs Assumptions, 4

- The size of the allowed electron orbits is

determined by a condition imposed on the

electrons orbital angular momentum - The allowed orbits are those for which the

electrons orbital angular momentum about the

nucleus is quantized and equal to an integral

multiple of h - h h / 2p

Mathematics of Bohrs Assumptions and Results

- Electrons orbital angular momentum
- mevr nh where n 1, 2, 3,
- The total energy of the atom is
- The total energy can also be expressed as
- Note, the total energy is negative, indicating a

bound electron-proton system

Bohr Radius

- The radii of the Bohr orbits are quantized
- This shows that the radii of the allowed orbits

have discrete valuesthey are quantized - When n 1, the orbit has the smallest radius,

called the Bohr radius, ao - ao 0.0529 nm
- n is called a quantum number

Radii and Energy of Orbits

- A general expression for the radius of any orbit

in a hydrogen atom is - rn n2ao
- The energy of any orbit is
- This becomes
- En - 13.606 eV/ n2

Specific Energy Levels

- Only energies satisfying the previous equation

are allowed - The lowest energy state is called the ground

state - This corresponds to n 1 with E 13.606 eV
- The ionization energy is the energy needed to

completely remove the electron from the ground

state in the atom - The ionization energy for hydrogen is 13.6 eV

Energy Level Diagram

- Quantum numbers are given on the left and

energies on the right - The uppermost level,
- E 0, represents the state for which the

electron is removed from the atom

Frequency of Emitted Photons

- The frequency of the photon emitted when the

electron makes a transition from an outer orbit

to an inner orbit is - It is convenient to look at the wavelength instead

Wavelength of Emitted Photons

- The wavelengths are found by
- The value of RH from Bohrs analysis is in

excellent agreement with the experimental value

Extension to Other Atoms

- Bohr extended his model for hydrogen to other

elements in which all but one electron had been

removed - Bohr showed many lines observed in the Sun and

several other stars could not be due to hydrogen - They were correctly predicted by his theory if

attributed to singly ionized helium

Orbits

- As a spacecraft fires its engines, the exhausted

fuel can be seen as doing work on the

spacecraft-Earth orbit - Therefore, the system will have a higher energy
- The spacecraft cannot be in a higher circular

orbit, so it must have an elliptical orbit

Orbits, cont.

- Larger amounts of energy will move the spacecraft

into orbits with larger semimajor axes - If the energy becomes positive, the spacecraft

will escape from the earth - It will go into a hyperbolic path that will not

bring it back to the earth

Orbits, Final

- The spacecraft in orbit around the earth can be

considered to be in a circular orbit around the

sun - Small perturbations will occur
- These correspond to its motion around the earth
- These are small compared with the radius of the

orbit