4. Homework #9
A nearby star is found to have a parallax angle of 40 arcsec, How far away is it in meters?
The flux of radiation we detect here on Earth for this star is 7x1010 W/m What is its Luminosity?
Where would we have to put this star so that it would have the same Luminosity as the Sun?

5. Homework #9 continued
A star has a photosphere whose temperature is 5772 K, what is the wavelength of the peak of its blackbody spectrum? What color is it?
If the star has a measured brightness of What is the luminosity of this star which is found to be 20 light years away from us?

6. How does light tell us the speed of a distant object?
This figure from the book can give an introduction to the Doppler effect.
The Doppler Effect

7. Explaining the Doppler Effect
Understanding the Cause of the Doppler Effect

8. Same for light
The Doppler Effect for Visible Light

9. Measuring the Shift
Stationary
Moving Away
Away Faster
Moving Toward
Toward Faster
We generally measure the Doppler effect from shifts in the wavelengths of spectral lines.

10. The amount of blue or red shift tells us an object’s speed toward or away from us:
The Doppler Shift of an Emission-Line Spectrum

11. Doppler shift tells us ONLY about the part of an object’s motion toward or away from us.
How a Star's Motion Causes the Doppler Effect

12. The Bizarre Stellar Graveyard
The objects depend on which force is resisting the crush of gravity:
Thermal pressure – main sequence stars and red giants
Electron degeneracy pressure – white dwarfs
Neutron degeneracy pressure – neutron stars
Black holes – gravity wins!!!

13. Electron Degeneracy White Dwarf
The central star collapses, heats up, and ejects a Planetary Nebula.
The star has insufficient mass to get hot enough to fuse Carbon.
Gravity is finally stopped by the force of electron degeneracy pressure.
The star is now stable…...
White Dwarf

14. White Dwarfs They generate no new energy.
They slide down the HR-diagram as they radiate their heat into space, getting cooler and fainter.
They are very dense; M packed into a sphere the size of the Earth!

15. White Dwarfs Degenerate matter obeys different laws of physics.
The more mass the star has, the smaller the star becomes!
increased gravity makes the star denser
greater density increases degeneracy pressure to balance gravity
1 M White Dwarf M White Dwarf

16. White Dwarfs Sirius B is the closest white dwarf to us
Sirius A + B in X-rays

17. A white dwarf is about the same size as Earth

18. When electron speeds in White Dwarf approach speed of light,
The White Dwarf Limit
Einstein’s theory of relativity says that nothing can move faster than light
When electron speeds in White
Dwarf approach speed of light,
electron degeneracy pressure
can no longer work since
electrons cannot go faster than
c – the speed of light
Chandrasekhar found that this
happens when a white dwarf’s
mass reaches 1.4 M

19. Calculated values for the limit will depend on the nuclear composition of the mass. Chandrasekhar got his solution based on the equation of state for an ideal Fermi gas
An equation of state is a thermodynamic equation which describes the state of matter under a given set of physical conditions. It provides a mathematical relationship between two or more state functions such as its temperature, pressure, volume, or internal energy

20. Gravitational constant
9/23/2017
Planck’s constant
Speed of light
Gravitational constant
Average molecular weight per electron (depends on chemical composition of star)
Mass of Hydrogen atom
~ from the Lane Emden equation
Equation of state in this case is a relation of pressure in an ideal fermi gas to other state variables

21. Poisson's equation is a partial differential equation
where is the Laplace operator, and and are real or complex-valued functions on a manifold.
When the manifold is Euclidean space, the Laplace operator is often denoted as
and so Poisson's equation is frequently written as
In Cartesian coordinates:

22. Newtonian gravity the gravitational field g is given by
9/23/2017
Newtonian gravity the gravitational field g is given by
Since the gravitational field is “well behaved” (conservative and irrotational) we can write it as a derivative of scalar:
Substituting into Gauss's law
Gauss’s law divergence of a vector field = source
yields Poisson's equation for gravity,
The so-called fundamental solution) is
which is equivalent to Newton's law of universal gravitation.

23. Why so far off?

24. This gives a value that’s a little high ~1. 7M
This gives a value that’s a little high ~1.7M. A more accurate value of the limit than that given by this simple model requires adjusting for various factors, including electrostatic interactions between the electrons and nuclei and effects caused by nonzero temperature. A rigorous derivation of the limit comes from a relativistic many-particle Schrödinger equation.
We will learn a little about this later…

25. Two Types of Supernova Massive star supernova:
Iron core of massive star reaches
white dwarf limit and collapses into a
neutron star, causing explosion. Contains
prominent hydrogen lines
White dwarf supernova:
Carbon fusion suddenly begins as white
dwarf in close binary system reaches white dwarf limit, causing total explosion. No
prominent lines of hydrogen seen.

26. Supernova Light Curves
(Type II)
(Type I)

27. The Bizarre Stellar Graveyard
The objects depend on which force is resisting the crush of gravity:
Thermal pressure – main sequence stars and red giants
Electron degeneracy pressure – white dwarfs
Neutron degeneracy pressure – neutron stars
Black holes – gravity wins!!!

28. Neutron Stars …are the leftover cores from supernova explosions.
If the core < 3 M, it will stop collapsing and be held up by neutron degeneracy pressure.
Neutron stars are very dense (1012 g/cm3 )
1.5 M with a diameter of 10 to 20 km
They rotate very rapidly: Period = 0.03 to 4 sec
Their magnetic fields are 1013 times stronger than Earth’s.
Chandra X-ray image of the neutron star left behind by a supernova observed in A.D. 386.
The remnant is known as G11.20.3.

29. Novae and Supernovae…
Hydrogen that accretes onto a neutron star builds up in a shell on the surface
When base of shell gets hot enough, hydrogen fusion suddenly begins leading to a nova

30. Nova explosion generates a burst of light lasting a few weeks and expels much of the accreted gas into space (H-bomb)

31. While a nova may reach about 100,000 L…
a white dwarf supernova attains 10,000,000,000 L (10 billion L)
runaway carbon fusion in core – causes explosion
since they all attain the same peak luminosity white dwarf supernovae make good distance indicators
they are more luminous than Cepheid variable stars so they can be used to measure out to greater distances than Cepheid variables

32. The cores of many Type II supernovae become neutron stars
When stars between 4 and 9 times the mass of the Sun explode as supernovae, their remnant cores are highly compressed clumps of neutrons called neutron stars.
These tiny stars are much smaller than planet Earth - in fact, are about the diameter of a large city (about 20 km)!!
suggested in 1934
max. mass of 3 solar masses (otherwise, collapse)
LGM’s discovered

33. LGM
In 1967, graduate student Jocelyn Bell and her advisor Anthony Hewish accidentally discovered a radio source in Vulpecula.
It was a sharp pulse which recurred every 1.3 sec.
They determined it was 300 pc away.
What was it?
Jocelyn Bell
LGM 1

34. A part of the LGM puzzle was solved when a pulsar was discovered in the heart of the Crab Nebula.
The Crab pulsar also pulses in visual light.

35. Pulsars

36. Some neutron stars in binary systems emit powerful jets of gas

37. Pulsars and Neutron Stars
Pulsars are the lighthouses of Galaxy!

38. Lighthouse Model

39. Pulsars and Neutron Stars
All pulsars are neutron stars, but all neutron stars are not pulsars!!
Synchotron emission --- non-thermal process where light is emitted by charged particles moving close to the speed of light around magnetic fields.
Emission (mostly radio) is concentrated at the magnetic poles and focused into a beam.
Whether we see a pulsar depends on the geometry.
if the polar beam sweeps by Earth’s direction once each rotation, the neutron star appears to be a pulsar
if the polar beam is always pointing toward or always pointing away from Earth, we do not see a pulsar

40. Rotation Periods of Neutron Stars
As a neutron star ages, it slows down.
The youngest pulsars have the shortest periods.
Sometimes a pulsar will suddenly speed up.
This is called a glitch!
There are some pulsars that have periods of several milliseconds.
they tend to be in binaries.
They are perhaps the most accurate clocks in the universe

41. Neutron Star