Binary Stars Astronomy 315 Professor Lee Carkner Lecture 9

Masses of Stars  While we can find the radius of a star from the Stefan-Boltzmann Law, we still do not know the mass  How do you find mass?  On Earth we weigh things  Weighing means measuring how gravity affects the object  How can we weigh things in space?  Watch how the star moves under the influence of the gravity of another star

Binary Stars  Most stars are in multiple systems  Our own sun is an exception  How do we find binary stars?  Some stars appear to be very close together on the sky  Called optical doubles  May just be a projection effect  We want stars that are gravitationally bound  In orbit around each other

Visual Binaries  The simplest type to observe are visual binaries  You can see one star orbit around another  The periods of such stars are often very long  Have to observe for decades to plot the orbit  Most visual binaries have a relatively stationary bright star and a moving fainter star

Binary Motion of Castor

Problems with Binaries  Period and Separation  In order to resolve the stars they have to have a large separation, but his also means a long period  Inclination  The orbit is not exactly face on to you, so you see its projection onto the plane of the sky

Inclination Effects

Using Binary Stars  What can we measure?  Orbital period  The time for one complete orbit  Orbital radius  The distance from each star to the center of mass  Need the distance to the binary from parallax first  What do we do with this information?  Need to understand gravity

Kepler’s Laws  In the early 1600’s Johannes Kepler published his laws of planetary motion  His first laws states that planetary orbits are elliptical  The longest axis of the ellipse is called the major axis (1/2 of it is called the semi-major axis a)  His third law states that the period (P) of the planet’s orbit (in years) squared is equal to the semi-major axis in astronomical units (AU) cubed (1 AU is the Earth-Sun distance) P 2 = a 3

Kepler’s Laws

Kepler and Newton  Kepler did not know why his laws worked  In the late 1600’s Isaac Newton used Kepler’s laws to develop his theory of gravity  The orbits of planets obey the Law of Universal gravitation  Gravitational force depends on mass  We can use Newton and Kepler’s laws together to find the mass of binary stars

Finding Masses  We can write a version of Kepler’s third law for binary stars: M A + M B = a 3 /P 2  where:  M A + M B is the combined mass of both stars in solar masses (M sun )  a is the semi-major axis of the orbit in astronomical units (AU)  P is the period of the orbit in years (yr)

Problems with Mass Determination  Our formula only gives us the sum of the masses  However, we can find the ratio of the masses by finding the distance to the center of mass for each star  Examples:  If one star is basically stationary, it has all the mass (like the sun and earth)  If both stars are equally distant from the center of mass they have the same mass  Ratio of mass is inverse ratio of distance to center of mass

Center of Mass Distances

Spectroscopic Binaries  There are very few visible binaries in the sky, so we have very few mass measurements  We have to try and find binaries in other ways  Easier to find double line spectroscopic binaries  We can’t resolve two individual stars (they are too close together)  however, we see two sets of spectral lines

Spectroscopic Binary Motion  What information can we get about the orbit if we can’t see it?  Can get the velocity of the orbit from the Doppler shift  More shifted the lines the faster the star is moving in its orbit  Can also get the period of the star from the Doppler shift  Time for Doppler shift to go from zero to max away to zero to max towards to zero

Spectroscopic Binary in Action

Velocities of Binary Components

Spectroscopic Binary Masses  The big problem with spectroscopic binaries is we do not know the inclination  Velocities highest in edge-on system and go to zero in face-on system  We only see component of Doppler shift for motion towards and away from us  We can get masses of stars statistically  Assume a random distribution of inclinations

Masses of Stars  Compare mass to position on HR diagram  Main sequence:  Cool, dim stars (red dwarfs) have low mass (M ~ M sun )  Medium-bright yellow stars have solar masses (M ~ M sun )  Hot, bright stars have high mass (M ~ 2-40 M sun )  White dwarfs  Mass about equal to sun  Giants  Large range of masses

Masses on the HR Diagram

Mass Distribution  There is a relationship between mass and luminosity for main sequence stars: L = M 3.5  Large mass. Large luminosity  White dwarfs are very dense  Solar mass in object the size of the Earth  Giants have low density  Generally much larger than main sequence stars of the same mass

Next Time  No homework Monday  First quiz on Monday  Covers all material since start of course through today  Multiple choice and short essay  Short essay include both written and problems  Be able to solve problems like the exercises and be able to write a paragraph explanation of key concepts  Study notes, exercises and readings  Study hard