Measuring the Properties of Stars

1. Measuring the Properties of Stars © Sierra College Astronomy Department

2. Measuring the Properties of StarsStellar Brightness and Luminosity • Power is the rate at which energy is transferred, or the amount of energy transferred per unit time. • Luminosity is the rate at which electromagnetic energy is being emitted - the total amount of power emitted by a star over all wavelengths. • Brightness (or apparent brightness) refers to the luminosity/area of a star as seen at the Earth and is related to the star’s luminosity through the inverse square law. © Sierra College Astronomy Department

3. Measuring the Properties of StarsStellar Brightness and Magnitude • In the second century B.C., Hipparchus created the first star catalog with corresponding brightnesses determined visually. • Hipparchus quantified each star’s brightness with a “magnitude”, an integer from 1 (brightest) to 6 (dimmest). • Ptolemy (second century A.D.) expanded the number of stars with measured magnitudes and popularized this system of measurement. • Today this magnitude is formally known asapparent magnitude and is designated by the letter m. © Sierra College Astronomy Department

4. Measuring the Properties of StarsStellar Brightness and Magnitude • The modern apparent magnitude scale is set up so that a 5-magnitude difference (say between two stars) is equal to a brightness change of 100 times (so as to closely match Hipparchus’s original data). • Consequently, a one-magnitude difference is equal to a brightness change of 2.512 times (2.5125 = 100). • The magnitude system is useful for its historical connections (for comparisons) and its manageable range (brightness itself covers a much larger range). © Sierra College Astronomy Department

5. Measuring the Properties of StarsStellar Brightness and Magnitude • Modern measuring devices allow astronomers to determine magnitudes to an accuracy of 0.001 or better. • Modern, large telescopes equipped with CCD devices can image objects as dim as 25th magnitude or better. • A few stars (e.g., Sirius) are so bright that they have negative magnitudes. Sirius’s apparent magnitude is –1.47. © Sierra College Astronomy Department

6. Measuring the Properties of StarsDistances to Stars - Parallax • Stellar parallaxes were not observed until the mid-1800s. • Parallax angle is half the maximum angle that a star appears to be displaced due to the Earth’s motion around the Sun. • The maximum angle of the nearest star is only about 1.52 seconds of arc, but astronomers define the parallax angle as half that value, or 0.76 seconds. © Sierra College Astronomy Department

7. Measuring the Properties of StarsDistances to Stars - Parallax • Parallax distance formula (in light-years): • Distance to star (ly) = • 3.26 ly/parallax angle in arcsec • Parallax distance formula (in parsecs): • Distance to star (pc) = • 1/parallax angle in arcsec • A parsec is the distance of a 1 AU object has a parallax angle of one arcsecond. One parsec is equal to 3.26 ly or 206,265 AU. © Sierra College Astronomy Department

8. Measuring the Properties of StarsDistances to Stars - Parallax • The satellite Hipparchos (1989-1993) measured parallax angles with very high precision (milli-arcsecond) for over 100,000 stars establishing highly accurate distance measurements out to about 1000 light-years. • Accurate stellar distances help to determine other quantities about celestial objects. © Sierra College Astronomy Department

9. Measuring the Properties of StarsAbsolute Magnitude & Luminosity • The intrinsic luminosity of a star is usually given as its absolute magnitudeand designated with a capitalM. • M is defined as the apparent magnitude a star would have if it were at a distance of 10 parsecs. • Sirius’s apparent brightness (–1.47) is due to its closeness (2.7 parsecs from Earth). Its absolute magnitude is +1.45 (determined by using inverse square law). © Sierra College Astronomy Department

10. Measuring the Properties of StarsAbsolute Magnitude & Luminosity • Given the brightness-magnitude relationship: • And the brightness-luminosity relationship: • It is possible to show derive the distancemodulus relationship (d is in parsecs): © Sierra College Astronomy Department

11. Measuring the Properties of StarsTemperature and Spectral Classes • A star’s color is determined by its temperature. • An absorption spectrum - the absorption of radiation at various wavelengths - can be used to determine a star’s temperature. • Harvard astronomers, lead by Edward Pickering and his women “computers” developed the first stellar classification system using letters A-O, in alphabetical order. • In particular, Williamina Fleming based the system on the strength of the stars’ hydrogen absorption lines (A strong, O weak) © Sierra College Astronomy Department

12. Measuring the Properties of StarsTemperature and Spectral Classes • The A-O scheme was eventually found to be inadequate. • Another “computer”, Annie Jump Cannon, discovered that a reordering and elimination of some of the letters gave a better scheme, which is still used today. • Cannon’s system was thought to reflect stellar composition, but “computer” Cecilia Payne-Gaposchkin showed that the system was a consequence of the stars’ surface temperatures. © Sierra College Astronomy Department

13. Measuring the Properties of StarsTemperature and Spectral Classes • The principal spectral types used today (from hottest to coolest) are designated as O B A F G K M. • O stars range in temperature from 30,000 K to 60,000 K. M stars have temperatures less than 3,500 K. • Within each spectral class, stars are subdivided into 10 categories by number (0 to 9). • For example, the Sun is a G2 star. • There are also other spectral types which are not quite as well known as the original seven (L and T types are two new ones used to classify very cool stars which form dust grains in their atmospheres). © Sierra College Astronomy Department

14. Measuring the Properties of StarsThe Hertzsprung-Russell Diagram • Hertzsprung-Russell diagram is a plot of absolute magnitude (or luminosity) versus temperature (or spectral class) for stars. • About 90% of all stars fall into a group running diagonally across the diagram called main-sequence stars. • Stars on the H-R diagram fall into categories such as main-sequence stars, white dwarfs, red giants, and supergiants. © Sierra College Astronomy Department

15. Measuring the Properties of StarsLuminosity Classes • In the 1880s Antonia Maury and Ejnar Hertzsprung discovered that the width of a star’s absorption lines was directly related to the star’s luminosity (which in turn is related to a star’s surface density). • Luminosity classes are one of several groups into which stars can be classified according to the characteristic widths of their spectra. • The luminosity classes are: Ia (supergiants), Ib (dimmer supergiants), II (bright giants), III (ordinary giants), IV (subgiants), and V (main-sequence). • Complete Stellar Classification: A star is fully classified if its spectral class and luminosity class are specified (e.g., the Sun is designated as a G2 V star) © Sierra College Astronomy Department

16. Measuring the Properties of StarsTowards a “Distance Ladder” Spectroscopic Parallax • Knowing a star’s luminosity class and temperature (spectral class) gives its absolute magnitude. • Knowing a star’s absolute magnitude and apparent magnitude gives its distance. • The distances to stars too far away for parallax measurements can be determined using this procedure. • Spectroscopic parallax represents the second rung (geometric parallax being the first) in the distance ladder created and used to scale the Universe. © Sierra College Astronomy Department

17. Measuring the Properties of StarsStar Sizes from Temperature and Luminosity The Sizes of Stars • The sizes of a few very large stars have been measured directly by interferometry. • Knowing the temperature of a star gives its energy emitted per square meter. • Knowing the total energy emitted (from the absolute magnitude) one can then calculate the surface area of the star. • From that the diameter of the star can be determined. © Sierra College Astronomy Department

18. Measuring the Properties of StarsMultiple Star Systems and Binaries Multiple Star Systems and Binaries • More than half of what appear as single stars are in fact multiple star systems. • Optical doubles are two stars that have small angular separation as seen from Earth but are not gravitationally linked. • Binary star system is a system of two stars that are gravitationally linked so that they orbit one another. © Sierra College Astronomy Department

19. Measuring the Properties of StarsMultiple Star Systems and Binaries • A visual binary is an orbiting pair of stars that can be resolved (normally with a telescope) as two stars. • If one uses large telescopes, about 10% of the stars in the sky are visual binaries. • Binaries can be confirmed by observing the system over time and looking for signs of revolution. • Spectroscopic binary is an orbiting pair of stars that can be distinguished as two due to the changing Doppler shifts in their spectra. © Sierra College Astronomy Department

20. Measuring the Properties of StarsMultiple Star Systems and Binaries • Algol, discovered by Goodricke in 1783, is an eclipsing binary in which one star moves in front of the other as viewed from Earth. • Algol’s light curve - a graph of the numerical measure of the light received from a star versus time - shows peaks and dips that indicate an unseen companion. © Sierra College Astronomy Department

21. Measuring the Properties of StarsMasses and Sizes from Binary Stars • Binary stars are important because they allow one to measure masses of stars using Newton’s version of Kepler’s laws. • Knowledge of the size of one of the star’s ellipses, along with knowledge of the period of its motion, permits calculation of the total mass of the two stars. • To determine how the system’s total mass is distributed between the two stars, one need only consider the ratio of the two stars’ distances to the center of mass. © Sierra College Astronomy Department

22. Measuring the Properties of StarsMasses and Sizes from Binary Stars • Because the inclination of spectroscopic binary orbits are usually not known, exact mass calculations cannot be done. • However, assuming an average inclination can provide information about average masses of spectroscopic binary stars. • Eclipsing binaries that are also spectroscopic binaries provide us with a way of measuring not only the masses of the two stars but also their sizes. • We derive this information using measurements of their Doppler shifts. © Sierra College Astronomy Department

23. Measuring the Properties of StarsThe Mass-Luminosity Relationship • Mass-luminosity diagram plots the mass versus the luminosity of a number of stars. • More massive stars are more luminous. • The mass-luminosity relationship holds only for main-sequence stars. where p has a value between 3.5 and 3.9 • The mass-luminosity relationship is valuable in investigating less accessible stars and in constructing and evaluating hypotheses on the life cycle of stars. © Sierra College Astronomy Department

24. Measuring the Properties of StarsMain-Sequence Lifetimes • The lifetime on the main-sequence depends on how much fuel (hydrogen) the star has and how fast the star is consuming it. • This lifetime can be expressed as: • Using the main-sequence mass-luminosity relation, we have: • where t⊙ is for the Sun and is ~ 10 billion years • Examples: A 10 M⊙ will last about 10 million years, whereas a 0.3 M⊙ star will last 300 billion years © Sierra College Astronomy Department

25. Measuring the Properties of StarsStar Clusters and Aging • Open (galactic) cluster is a group of stars that share a common origin and are located relatively close to one another. • Globular cluster is a spherical group of up to hundreds of thousands of stars found primarily in the halo of the Galaxy. • Clusters are important for two reasons: • All stars in a cluster are at about the same distance from us, so their apparent magnitude is a direct indication of their absolute magnitude. • All the stars within a cluster formed at about the same time (more or less). • Age of cluster determined from main-sequence turnoff • Much of our knowledge of star formation has come from examination of clusters © Sierra College Astronomy Department

26. Measuring the Properties of StarsVariable Stars as Distance Indicators • Not all stars shine steadily like the Sun. • Stars that vary significantly over time are called variable stars. • A certain sub-class of variable stars are called pulsating variable stars (based on how the star is pulsates in size) • Most pulsating variable stars occupy the instability strip on the H-R diagram. • A special class of very luminous pulsating variable stars called Cepheid variable stars have a well established period-luminosity relation that provides a powerful means for determining cosmic distances. © Sierra College Astronomy Department