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
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

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
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

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
  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

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
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

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