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## Gravity, Planetary Orbits, and

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Title: Gravity, Planetary Orbits, and

1
Chapter 11
• Gravity, Planetary Orbits, and
• the Hydrogen Atom

2
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

3
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

4
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

5
• 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

6
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

7
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

8
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

9
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

10
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

11
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

12
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

13
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

14
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

15
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

16
• 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

17
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

18
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

19
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

20
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

21
Keplers Second Law
• Is a consequence of conservation of angular
momentum
• The force produces no torque, so angular momentum
is conserved

22
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

23
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

24
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

25
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

26
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

27
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

28
Energy, cont.
• Since Ug goes to zero as r goes to infinity, the
total energy becomes

29
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

30
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

31
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

32
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

33
Escape Speed, General
• The Earths result can be extended to any planet
• The table at right gives some escape speeds from
various objects

34
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

35
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

36
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

37
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

38
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

39
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

40
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.

41
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

42
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

43
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

44
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

45
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

46
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

47
Emission Spectra Examples
48
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

49
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

50
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

51
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

52
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

53
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

54
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

55
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
• 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

56
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

57
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

58
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

59
• 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

60
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

61
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

62
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

63
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

64
Wavelength of Emitted Photons
• The wavelengths are found by
• The value of RH from Bohrs analysis is in
excellent agreement with the experimental value

65
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

66
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

67
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

68
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