Chapter 11

Gravity, Planetary Orbits, and

the Hydrogen Atom

Newton’s 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 Nm2 / 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 Newton’s

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

Active Figure AF_1101 gravitational force.swf

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

earth’s 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

earth’s 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 Earth’s

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

Kepler’s Laws, Introduction

Johannes Kepler was a German astronomer

He was Tycho Brahe’s assistant

Brahe was the last of the “naked eye” astronomers

Kepler analyzed Brahe’s data and formulated three laws of planetary motion

Kepler’s Laws Kepler’s First Law

Each planet in the Solar System moves in an elliptical orbit with the Sun at one focus

Kepler’s Second Law The radius vector drawn from the Sun to a planet

sweeps out equal areas in equal time intervals Kepler’s 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 < e < 1

Active Figure AF_1105 properties of ellipses.swf

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

Kepler’s 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

Halley’s comet has an orbit with high eccentricity (b) eHalley’s comet = 0.97

Speed and Eccentricity AF_1107 elliptical orbits.swf

Kepler’s Second Law

Is a consequence of conservation of angular momentum

The force produces no torque, so angular momentum is conserved

Kepler’s 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

Kepler’s 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

Kepler’s 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

Kepler’s 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

Kepler’s 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 >> 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 < 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 Earth’s 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 Earth’s 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 galaxy’s 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, , 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

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 Earth’s 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 hydrogen’s 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 Planck’s ideas of quantized energy levels to orbiting electrons

Bohr’s 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

Bohr’s 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 electron’s motion

This representation claims the centripetally accelerated electron does not emit energy and eventually spirals into the nucleus

Bohr’s 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 atom’s energy The frequency is independent of the frequency of the

electron’s orbital motion The frequency of the emitted radiation is given by Ei – Ef = hƒ h is Planck’s constant and equals 6.63 x 10-34 Js

Bohr’s Assumptions, 4 The size of the allowed electron orbits is

determined by a condition imposed on the electron’s orbital angular momentum

The allowed orbits are those for which the electron’s orbital angular momentum about the nucleus is quantized and equal to an integral multiple of h h = h / 2

Mathematics of Bohr’s Assumptions and Results Electron’s 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 values—they 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 Bohr’s 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