Star Properties - Notes

Star Properties

The Solar Neighborhood Properties of Stars

  • Topics covered include:
    • Luminosity and Apparent Brightness
    • Magnitude Scale
    • Stellar Temperatures
    • Spectral Class
    • Stellar Sizes (Radii)

Units

  • Topics covered include:
    • The Hertzsprung-Russell Diagram
    • Extending the Cosmic Distance Scale
    • Stellar Masses
    • Mass and Other Stellar Properties

Stellar Distances and Parallax

  • Stellar distances can be measured using parallax with the formula: d=1/pd = 1/p, where:
    • dd is the distance in parsecs (pc).
    • pp is the parallax in arc seconds.
  • 1 arc second = 1/36001/3600th of a degree.
  • 1 pc = 3.3 light-years (ly).
  • Parallax is the apparent motion of an object against a distant background from two vantage points and represents the first step on the distance ladder.

The Solar Neighborhood

  • Nearest star to the Sun: Proxima Centauri, part of the Alpha Centauri complex (a three-star system).
  • Model of distances:
    • Sun: Marble
    • Earth: Grain of sand orbiting 1 meter away
    • Nearest star: Another marble 270 km away
    • Solar system extends about 50 m from Sun; rest of distance to nearest star is basically empty.
  • The 30 closest stars to the Sun are listed with approximate distances of 1-3 pc

Barnard’s Star

  • Barnard’s Star (1.8 pc or 5.9 ly) has the largest proper motion.
  • Proper motion: actual shift of the star in the sky, corrected for parallax.

Properties of Stars

  • Intrinsic properties:
    • Mass
    • Radius
    • Luminosity
  • Other properties:
    • Distance
    • Brightness (absolute and apparent)
    • Proper Motion
    • Stellar Class
    • Temperature

Measuring the Stars

  • A table summarizes stellar properties, measurement techniques, known quantities, measured quantities, relevant theory, and corresponding sections.
  • Distance
    • Stellar parallax: astronomical unit, parallactic angle, elementary geometry.
    • Spectroscopic parallax: main sequence, spectral type, inverse-square law.
  • Radial velocity: speed of light, spectral lines, Doppler effect, atomic spectra.
  • Transverse velocity: distance, proper motion, elementary geometry.
  • Luminosity
    • distance, apparent magnitude, inverse-square law.
    • main sequence, spectral type.
  • Temperature
    • photometry: color, blackbody law.
    • spectroscopy: spectral type, atomic physics
  • Radius
    • direct: distance, angular size, elementary geometry.
    • indirect: luminosity, radius-luminosity-temperature relationship, temperature
  • Composition: spectrum, atomic physics.
  • Mass: observations of binary stars (distance), binary period, binary orbit, Newtonian gravity and dynamics, orbital velocity.

Luminosity and Apparent Brightness

  • Luminosity: total power radiated by a star (energy radiated per second), measured in Watts.
  • Luminosity or absolute brightness: how bright a star actually is (at 10 pc).
  • Apparent brightness: how bright a star appears due to its distance from Earth; depends on absolute brightness and distance.
  • Two stars can appear equally bright if a closer, dimmer star and a farther, brighter one are compared.
  • Apparent brightness is measured using the magnitude scale, also known as apparent magnitude (mm).
    • It's a logarithmic scale: a change of 5 in magnitude corresponds to a factor of 100 in apparent brightness.
    • Larger magnitudes are fainter; smaller magnitudes are brighter.
  • Luminosity is related to absolute brightness and is measured in magnitude, which is also a log scale. Absolute Magnitude (MM).
  • Converting from magnitude to luminosity in solar units: the Sun has an absolute magnitude of +5.

Inverse Square Law & Distance Modulus

  • Light follows an inverse square law:
    • Intensity decreases by the square of the distance from the light source.
  • Absolute Magnitude and apparent magnitude are related by distance.
  • Distance modulus equation:
    • m−M=5log⁡(d/10)m - M = 5 \log(d/10), where d is in parsecs.
    • m−Mm - M is the distance modulus.
    • If the distance is 10 pc, then M=mM = m, and the distance modulus is zero.

Stellar Temperatures

  • Star color indicates temperature.
    • Red stars are relatively cool, while blue stars are hotter.
  • Color provides a relative comparison, not an exact temperature.
  • Radiation from stars is blackbody radiation.
    • Observations at two wavelengths (blue and visual) can define the temperature because the blackbody curve isn't symmetric.
    • Wien’s Law can also be used.

Stellar Colors and Temperatures

  • A table shows stellar colors and temperatures. Values are approximate.
  • Columns show B flux, V flux, color, familiar examples, and surface temperature.
    • Blue: 30,000K, Rigel.
    • Blue-violet: 20,000K, Mintaka
    • White: 10,000K, Vega, Sirius
    • Yellow-white: 7000K, Canopus
    • Yellow: 6000K, Sun, Alpha Centauri
    • Orange: 4000K, Arcturus, Aldebaran
    • Red: 3000K, Betelgeuse, Barnard's Star.

Cecilia Payne – Gaposchkin (1900-1979)

  • Stellar spectra is more informative than blackbody curves.
  • Women at Harvard Observatory (1890s):
    • Henrietta S. Leavitt, Williamina Fleming, Cecilia Payne, and Annie Cannon.
  • Many women compiled data for the Henry Draper catalog.
  • Fleming started a classification of 22 classes (A-P using stellar spectra), rearranged by Cannon into order of decreasing surface temperature.
  • Seven general categories of stellar spectra (spectral class) exist, corresponding to different temperatures.
    • From highest to lowest temperature: O, B, A, F, G, K, M.
    • These categories are unrelated to star size.
  • In 1925, Payne was the first person to receive a Ph.D. in astronomy from Harvard.
  • Payne calculated a temperature scale to match Cannon's classification system.
  • She theorized that stars are mostly hydrogen.

Stellar Spectra

  • Spectral lines give composition.
  • Temperatures corresponding to OBAFGKM spectral types: 30,000K, 20,000K, 10,000K, 7000K, 6000K, 4000K, 3000K.
  • Elements and molecules include: Hydrogen, Helium, Carbon, Iron, Calcium, Sodium, Magnesium, Oxygen, and Many molecules.

Spectral Class

  • Examples are shown of the spectral classifications in a table, including approximate surface temperature K, noteworthy absorption line, and familiar examples
    • O: 30,000, Ionized helium strong, Mintaka
    • B: 20,000, Neutral helium moderate, Rigel
    • A: 10,000, Neutral helium very faint, Vega, Sirius
    • F: 7000, Singly ionized heavy elements, Canopus
    • G: 6000, Singly ionized heavy elements, Sun, Alpha Centauri
    • K: 4000, Singly ionized heavy elements, Arcturus, Aldebaran
    • M: 3000, Neutral atoms strong, Betelgeuse, Barnard's Star

Spectral Class Subdivisions

  • Each spectral class is subdivided into 10 subclasses using numbers 0-9 (e.g., G0, G1, …, K0, K1, …).
  • The Sun is a G2 spectral type star.
  • Also, stars have a luminosity class (e.g., V for main sequence), so the Sun's classification is G2V.

Stellar Sizes

  • Spectral class or temperature gives no indication of stellar size.
  • Examples of stars show that there is no correlation between star size and temperature.
    • Blue giant: 10 L☉, 20 R☉, 13,000 K.
    • Red giant: 80 L☉, 20 R☉, 4000 K.
    • Red dwarf: 0.05 L☉, 0.5 R☉, 4000 K.

Stellar Sizes

  • A few very large, very close stars can be imaged directly using interferometry.
  • Betelgeuse, spectral class M (cooler than G), is larger than the Sun.
  • Betelgeuse's radius is approximately 600 R☉.
  • The earth's orbit is compared to the stellar size.

Stellar Radii

  • Stellar radii vary widely:
    • Dwarf stars: radii equal to or less than the Sun’s.
    • Giant stars: radii between 10 and 100 times the Sun’s.
    • Supergiant stars: radii more than 100 times the Sun’s.

Calculating Stellar Size

  • For stars that cannot be imaged directly, size is calculated knowing luminosity and temperature:
    • L=R2T4L = R^2T^4 (if luminosity, radius, and temperature are measured in solar units).

The Hertzsprung-Russell Diagram

  • The H-R diagram plots stellar luminosity against surface temperature.
  • It is the most important diagram in astronomy.

H-R Diagram Patterns

  • When many stars are plotted on an H-R diagram, a pattern forms.
  • The 80 closest stars to us:
    • Dashed lines of constant radius.
    • The darkened curve is the main sequence, where stable stars burn hydrogen to helium.
    • Stars spend most of their life here.
    • The white dwarf region contains hot but not very luminous stars (small).

H-R Diagram - 100 Brightest Stars

  • An H-R diagram of the 100 brightest stars looks quite different.
  • These stars are all more luminous than the Sun.
  • Two new categories appear here:
    • Red giants
    • Blue giants
  • The brightest stars in the sky appear bright because of their enormous luminosities, not their proximity.

H-R Plot of 20,000 Stars

  • This is an H-R plot of about 20,000 stars within a few hundred parsecs of the Sun.
  • The main sequence is clear, as is the red giant region.
  • The luminosity class and spectral type are plotted to visualize the stars within 30,000-3,000K

Luminosity Class

  • A diagram shows luminosity class designation and spectral classification on the H-R diagram
  • Ia: Brightest Supergiants
  • Ib: Supergiants
  • II: Bright giants
  • III. Normal giants
  • IV: Subgiants
  • V: Main sequence stars

Stellar Luminosity Classes

  • Luminosity classes are described:
    • Ia: Bright supergiants
    • Ib: Supergiants
    • II: Bright giants
    • III: Giants
    • IV: Subgiants
    • V: Main-sequence stars and dwarfs

Line Width as Spectral Luminosity

  • Width of absorption spectral lines is used to classify luminosity for stars.
  • Stars can have the same spectral type but different luminosity classes.

Giants vs Main Sequence

  • Giants and supergiants can be distinguished from main-sequence stars.
  • A table illustrates variation in stellar properties within a spectral class:
    • K2V main-sequence star (ε Eridani): 4900 K, 0.3 L☉, 0.8 R☉.
    • K2III red giant (Arcturus): 4500 K, 110 L☉, 21 R☉.
    • K2Ib red supergiant (ε Pegasi): 4300 K, 4000 L☉, 140 R☉.

Spectroscopic Parallax

  • Spectroscopic parallax can extend the cosmic distance scale to several thousand parsecs.
  • Uses the H-R diagram.

Spectroscopic Parallax Technique.

  • Has nothing to do with parallax, but does use spectroscopy in finding the distance to a star.
    1. Determine Temperature and spectral class
    2. Use spectral line width to determine luminosity class
    3. Place on H-R diagram
    4. Infer Luminosity from H-R diagram (y-axis)
    5. Calculate distance from distance modulus equation
  • The spectroscopic parallax calculation can be misleading if the star is not on the main sequence which may give an incorrect value for distance.

Stellar Masses

  • Many stars are in binary pairs; measurement of their orbital motion allows determination of the masses of the stars.
  • Visual binaries can be measured directly (e.g., Kruger 60).
  • Newton’s modification of Kepler’s Third law applies:
    • M1+M2=A3/P2M1 + M2 = A^3 / P^2

Stellar Masses

  • Spectroscopic binaries use the spectral lines to give the motion of the stars relative to each other using the Doppler shift.

Eclipsing Binaries

  • Eclipsing binaries can be measured using the changes in luminosity.

Stellar Mass

  • Mass is the main determinant of where and how long a star will be on the Main Sequence and how long it will live.

Mass Distribution

  • This pie chart shows the distribution of stellar masses.
  • The more massive stars are much rarer than the least massive.
  • Red dwarfs are the most common star and live the longest.

Mass Correlation

  • Mass is correlated with radius and is very strongly correlated with luminosity:
    • As M increases, R increases
    • As R increases, L increases

Mass and Stellar Lifetime

  • Mass is also related to stellar lifetime:
    • stellar lifetime ∝ ML\frac{M}{L}
  • Using the mass-luminosity relationship:
    • stellar lifetime ∝ 1M3.5\frac{1}{M^{3.5}}

Stellar Lifetimes

  • The most massive stars have the shortest lifetimes because they burn their fuel at a very rapid pace.
  • Small red dwarfs burn their fuel extremely slowly and can have lifetimes of a trillion years or more.
  • Examples:
    • Sun - 10 billion years
    • Star 9 times the mass of the Sun – 10 million years

Key Properties of Main-Sequence Stars

  • A table shows key properties of some well-known main-sequence stars:
    • Star, Spectral Type, Mass (Solar Masses), Central Temperature (10^6 K), Luminosity (Solar Luminosities), Estimated Lifetime (M/L) (10^6 years)
    • Spica B: B2V, 6.8, 25, 800, 90
    • Vega: A0V, 2.6, 21, 50, 500
    • Sirius A: A1V, 2.1, 20, 22, 1000
    • Alpha Centauri: G2V, 1.1, 17, 1.6, 7000
    • Sun: G2V, 1.0, 15, 1.0, 10,000
    • Proxima Centauri: M5V, 0.1, 0.6, 0.00006, 16,000,000

Summary

  • Can measure distances to nearby stars using parallax.
  • Apparent magnitude is related to apparent brightness.
  • Absolute magnitude is a measure of the power output of the star (L).
  • Spectral analysis gives surface temperature of stars and has led to the defining of seven spectral classes of stars.
  • Each category of spectral class can be subdivided into 10 subclasses.
  • Stellar radii can be calculated if distance and luminosity are known.
  • The H-R diagram is a useful tool in astronomy.

Summary (cont.)

  • Luminosity class can distinguish giant star from main-sequence one in the same spectral class.
  • Spectroscopic parallax is a new tool use to find distance using the H-R diagram.
  • Measurements of binary-star systems allow stellar masses to be measured directly.
  • Mass is well correlated with radius and luminosity.
  • Stellar lifetimes depend on mass; the more the mass, the shorter the lifetime.