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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 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 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
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


Kepler’s Second Law


Each planet in the Solar System moves in an
elliptical orbit with the Sun at one focus
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
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
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

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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, 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 Earth’s atmosphere
Balmer Series

In 1885, Johann Balmer found an
empirical equation that correctly
predicted the four visible emission lines
of hydrogen

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H is red, l = 656.3 nm
H is green, l = 486.1 nm
H is blue, l = 434.1 nm
H is violet, l = 410.2 nm
Emission Spectrum of
Hydrogen – Equation

The wavelengths of hydrogen’s spectral lines
can be found from

RH is the Rydberg constant

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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

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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


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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

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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 / 2p
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