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/m2 What is its Luminosity? Where would we have to put this star so that it would have the same Luminosity as the Sun?

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?

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

Explaining the Doppler Effect Understanding the Cause of the Doppler Effect

Same for light The Doppler Effect for Visible Light

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.

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

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

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!!!

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

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; 0.5 - 1.4 M packed into a sphere the size of the Earth!

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 1.3 M White Dwarf

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

A white dwarf is about the same size as Earth

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

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

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 ~ 2.018236 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

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:

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.

Why so far off?

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…

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.

Supernova Light Curves (Type II) (Type I)

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!!!

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.

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

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

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

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

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

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.

Pulsars

Some neutron stars in binary systems emit powerful jets of gas

Pulsars and Neutron Stars Pulsars are the lighthouses of Galaxy!

Lighthouse Model

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

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

Neutron Star