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Measuring the Properties of Stars. Measuring the Properties of Stars Stellar Brightness and Luminosity. Power is the rate at which energy is transferred, or the amount of energy transferred per unit time.

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Measuring the Properties of Stars

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Measuring the properties of stars l.jpg

Measuring the Properties of Stars

© Sierra College Astronomy Department


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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