How To Determine The Size Of A Star

Measuring the Stars
Giants, Dwarfs, and the Principal Sequence

October xvi, 2016
Astronomy 122
Prof. J. Brau

Outline

• The Distances to the Stars
  • (see also Extending the Cosmic Distance Scale)
• Stellar Motion
• Stellar Sizes
• Luminosity and Brightness
• Temperature and Color
• The Classification of Stars
• The Hertzsprung-Russell Diagram
• Extending the Cosmic Distance Calibration
• Luminosity Classes
• Stellar Mass
• Stellar Lifetime
• Central Backdrop of Some Well-Known Stars
• Star Clusters
• Summary of Star Measures

---

The Distances to the Stars

---

Stellar Motion

Stars are moving and the motility across the sky, subsequently correction for parallax, is called proper motion The largest known proper movement of any star is that of Barnard'southward star (227 arc-seconds in 22 years) The true space motion is the combination of the transverse (proper) move and the radial motion, adamant from the Doppler shift of the stellar lines.

---

Stellar Sizes

Most stars announced as points of lite, so their sizes cannot exist straight measured Even so, for a few we measure the size direct: Betelgeuse Betelgeuse is located in the constellation Orion for well-nigh we must use indirect means (temperature and brightness) Consider a "hot star" with surface temperature 30,000 K, and a "cool star" with surface temperature 3,000 K place these stars from the peak of their blackbody curves These stars take names: Type O : thirty,000 K Type 1000 : 3,000 G

• What are their luminosities?
  • We meet this from a plot of the "main sequence" of the Hertzsprung-Russell Diagram
  • So the Type O star is 10¹⁰ times every bit bright every bit a Type M star
• How does this compare to our expectations from Blackbody Theory?
  • Stefan-Boltzmann Law says the energy emitted per unit of measurement area is proportional to T^iv
    or emission per unit surface area from type O (30,000K) should exceed type M (iii,000K) by:
    10⁴ = 10,000
    only not past ten¹⁰ = x,000,000,000 that we observe
    we have an extra factor of 1,000,000 = 10^half-dozen

  So why are type O stars and so much brighter? Because blazon O are as well much bigger Total brightness is proportional to Area x T4, not but T4 or Luminosity ~ (Radius) 2 x Tiv So the extra factor of 1,000,000 = tenhalf-dozen represents an increased expanse, or a radius increment of i,000 that is, type O has a radius 1000 times the radius of blazon M The to a higher place equation can be rearranged to derive an equation for the radius: (Radius)two ~ Luminosity / T four or Radius ~ SQRT(Luminosity) / T 2 look once again at the radius trends on the plot of the "chief sequence" of the Hertzsprung-Russell Diagram Stellar Sizes Stars tin can be classified by their size Giants (ten-100 times the Dominicus'due south radius) Supergiants (100-yard times the Sun'due south radius) Dwarf (comparable to or smaller than the Sunday)

---

Luminosity and Effulgence

Luminosity:

Total rate at which radiative energy is given off by a celestial body




Effulgence The effulgence that the star appears to have to an observer on the Earth This depends on how far away the object is by the changed-foursquare police: Knowing Brightness and Altitude, we can make up one's mind Luminosity Brightness ~ luminosity / distance 2 Often expressed relative to the lord's day's luminosity (LSUN). Magnitude: Measure of Effulgence Apparent magnitude: Greeks (Hipparchus) established scale called Credible magnitude Credible magnitude is a measure out of Effulgence Brightest stars visible to unaided eye = Magnitude 1 Dimmest stars visible to unaided center = Magnitude vi This is a logarithmic calibration Measurements testify 1st magnitude stars are 100x as brilliant as 6th magnitude stars. So, a Magnitude deviation of ane corresponds to a gene of 2.51 in effulgence. or (two.51)5 = 100



Absolute Magnitude:
Apparent magnitude a star would accept if it were exactly x parsecs from the Earth.
Accented Magnitude = Luminosity, although in different units.

Luminosity and Brightness of the Sun

• Sun'southward Brightness = 1370 Watts/yardⁱⁱ
• Sun's Distance (d) = one.5 10 10^elevenm
• Therefore, Area of Sphere is 4 (pi) d^two
  = iv (pi) (1.5 10 10^elevenm)²
  = 3 10 10²³ m²
• Luminosity= (1370 Watts/m²)(three x x²³ grand²)
  = iv ten 10²⁶ Watts

  At a distance of 10 parsecs the Sun would be a magnitude iv.83 star
  So the Absolute Magnitude of the Sunday is four.83

Another example: Sirius:

Brightest Star in the Heaven

• Credible Magnitude = -1.44
• Distance
  • parallax = 0.38 arcsec
  • distance (d) = i/parallax = ii.5 parsec
• 3.26 ly in a parsec
• So, d = 8.6 ly (=two.5 x 3.26)
  
• Luminosity= 22 x L_{lord's day}
  and Absolute Magnitude is ane.45

  equally shown in the textbook
  

  Sun has absolute magnitude four.83
  Sirius has absolute magnitude 1.45
  

  So
  two.51^iv.83-1.45 = 22
  or Sirius has 22 times the Luminosity of the Sunday

---

Temperature and Color

Star'south Surface Temperature Our understanding of Blackbody Radiations explains why star's colour is related to surface temperature. Colors of Betelgeuse and Rigel in Orion are clearly cerise and blue B (blueish) and 5 (visual) filters admit different amounts of light for different temperatures The color alphabetize, relating the B and Five intensities, is defined in two ways ratio of B to V (B/5) deviation between B and 5 (B-Five) relationship between color index and temperature a blood-red star has a surface temperature of about 3,000 K a blue star has a surface temperature of about xx,000 K Star's absorption spectrum is also indicative of the surface temperature. Ionization land of atoms depends on temperature Energy of light (and therefore absorption) depends on temperature

---

The Nomenclature of Stars

Star's Absorption Spectrum
Used to allocate Stars (according to Spectral Class)



Mnemonic for O B A F G Chiliad K :

Oh, Be A Fine Guy (Daughter), Kiss Me



Types of Spectra

Hydrogen Lines Strongest in A spectra

Molecular Lines Strongest in M spectra

Neutral Metals Strongest in G,K, and One thousand

Neutral Helium Strongest in B

Ionized Helium Strongest in O


The spectral class matches with surface temperature

Each lettered spectral class is further sub-divided in x subdivisions, denoted by 0-9

• so, for example:
  • The Dominicus is G2,
  • Betelgeuse is M2,
  • and Barnard'southward star is M5

---

The Hertzsprung-Russell Diagram

---

Extending the Cosmic Distance Scale

---

Luminosity Classes




Merely distances to stars on the main-sequence can be correctly determined past the simple procedure outlined above under spectroscopic parallax since nine out of 10 stars are on the main-sequence, this is already pretty skilful Can we distinguish stars that are off the primary-sequence by their spectra? Yes Technical Deviation in Spectra of Aforementioned Spectral Class based on the width of lines width of lines determine luminosity classes Ia-5


So at present we have a third characterization for a star:
• The Dominicus is G2V, 5 beingness the stellar luminosity grade for the main-sequence.
• Betelgeuse is M2Ia

An case of the discrimination between K2-type stars is illustrated in the following table:

---

Stellar Mass

Stellar Masses and Binary Stars


Most stars are office of a multiple star organization Visual Binary - Run across Orbiting Stars - double star Kruger lx Spectroscopic Binary - Spectrum reveals binary nature thru Doppler Event Eclipsing Binary - Lite curve shape From these measurements we tin learn about masses: For visual binary use Kepler's law to deduce masses: M1 + M2 ~ a3 /p2 a = semimajor axis = radius for circular orbit p = menstruum of orbit Annotation: I don't look you lot to know this formula, just that there is a human relationship between the mass, the flow, and the size (semimajor axis) of the orbit For other binaries, some information on orbits and masses can exist derived Nigh of our cognition of the masses of stars is based on these binary measurements Example binary star: Sirius (the brightest star in the heaven) vivid Sirius A and faint companion Sirius B orbital period = 50 years semi-major centrality = 20 AU GrandA + MB = 3.ii Chiliadsun farther study reveals: MA = two.i Mlord's day MB = 1.1 Grandsunday


Stellar properites depend on the mass

• variation of mass forth the principal sequence


The mass of the star at the time of germination determines its location on the main sequence.

Master sequence stars range in mass from 0.1 to 20 times the mass of the Sunday (with a few exceptions)

Near principal-sequence stars are low-mass stars, and only a small fraction are much more massive than the Lord's day

The main-sequence star's radius and luminosity depend on its mass

• mass-radius relation for main sequence stars
• mass-luminosity relation for principal sequence stars
  • luminosity ~ mass³ or mass⁴ (judge)

---

Stellar Lifetimes

• Luminosity increases as (Mass) ^three for massive main-sequence stars
  and (Mass) ⁴ for more mutual main-sequence stars
  
  reference to details by Nick Strobel
• Full fuel to fire in star is the mass
• Therefore:
  ............. More massive stars fire up fastest and have shortest lives
  • 

    and since the luminosity increases equally the cube of the mass for the almost massive stars:

  • 

    (these are only approximate relationships)

---

Central Properties of Some Well-Known Stars



H-R diagram of some well-known stars

---

Star Clusters

Star clusters are a collection of stars at approximately the aforementioned altitude from u.s.a., so they can be compared without correcting the brightness to absolute brightness (or we don't need to know how far away they are to compare them)

Open (or Galactic) Clusters a loose association of young stars - metal-rich Example: The Pleiades: we assume these stars were formed around the same time O type stars are withal on the main-sequence, therefore this cluster is younger than lifetime of O type stars: 25 million years heavy elements are abundant in these stars metallic-rich created by earlier stars

H-R Diagram for the Pleiades




Globular Clusters large spherical clusters of stars- metal-poor Instance: Omega Centauri again, we presume these stars were formed around the same time no O or B type stars no main-sequence stars with masses greater than 0.8 solar masses A type stars are passing back through the main-sequence few heavy elements are institute in these stars metal-poor so stars were created in afar by, when heavy elements were less abundant We can conclude that this (and other) globular cluster was formed over x billion years ago recall lifetime of the Sun volition be 10 billion years, and if information technology had formed with this cluster is would have been extinguished by now, every bit seen on the H-R diagram

H-R Diagram for Omega Centauri



• Open clusters are of recent (last billion years) origin
• Globular clusters are very old (more than ten billion years)

---

Summary of Star Measures

---

These lecture notes were developed for Astronomy 122 by Professor James Brau, who holds the copyright. They are made bachelor for personal apply past students of the course and may not be distributed or reproduced for commercial purposes without my express written consent.

How To Determine The Size Of A Star,

Source: https://pages.uoregon.edu/jimbrau/astr122/Notes/Chapter17.html

Posted by: rodriguezplad1987.blogspot.com

Share this post