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Measuring stars (Stars change (variable stars # (pulsating variables…

- - - - observed a supernova #
        
        trigonometric parallax
        
        Showed supernova lay beyond moon
  - - - intrinsic to star
      - arcsec per year
      - fastest proper motion shown by Barnard's Star
      - Rigel smallest
      - transverse velocity
        
        speed of motion across our line of sight
        
        :bulb: Transverse speed (magnitude of transverse velocity) is given by: \( v_t=d×(\mu/\text {radians}\)
        
        A smaller proper motion at a longer distance corresponds to a larger transverse speed than the same proper motion at a shorter distance
        
        \[d \text{ is distance and }\mu \text{ is proper motion.}\]
      - radial velocity
        
        star's motion directly towards or away from us
        
        can be measured by Doppler effect
        
        redshift: \(\lambda\) increases
        
        blueshift: \(\lambda\) decreases
        
        :bulb:\[v_r=c\times (\lambda '-\lambda)/\lambda \]
        
        \( \lambda ' \) is the observed wavelength and c is the speed of light
    - - :bulb: given by \(v=\sqrt{v_t^2+v_r^2}\)
  - - - binary stars #
        
        two stars, a system
        
        their angular separations so small that they cannot be distinguished
        
        visual binary systems
        
        both seen as distinct points of light
        
        Sirius
        
        component stars referred to by name and letter (e.g. Sirius A)
        
        useful in determining mass of the component stars
        
        Circular orbits
        
        centre of mass
        
        speeds of the two stars constant, but they are accelerating because direction is always changing
        
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        if you could sit on M and observe m
        
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        lies at centre of each of the circular orbits of the two stars
        
        Elliptical orbits
        
        foci
        
        the further apart they are, the more eccentric the orbit is
        
        1 more item...
        
        length given by semimajor axis a
        
        half the longest axis
        
        :bulb: Period of ellipses given by \[P=2\pi\left(\frac{a^3}{G(M+m)}\right)^{1/2}\]
        
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        orbits usually seen at an angle to Earth and not straight in front of us
        
        usually viewed as a drawing on a draughtsman's table rather than a drawing hanging on the wall
        
        the semi-major axis looks to be a different length at an angle
        
        The projected semimajor axis, a', can be found using the angular separation of the two stars, \(\phi\),
        
        :bulb: \[a'=(\phi / \text {radians}) \times d\]
        
        spectroscopic binaries
        
        can be measured by measuring Doppler shift
        
        must be bright enough to give good quality spectra
        
        must have radial velocity component
        
        can't be at right angle to line of sight
        
        eclipsing binaries
        
        we view then edge on
        
        one star is seen to pass in front of the other
        
        dip in brightness
        
        rare
        
        The Herschels
        
        size and speed calculations from non-visual binaries (stereoscopic and eclipsing)
        
        :bulb: \[M/m=d_m/d_M=v_m/v_M\]
        
        :bulb:\[v_M=2\pi d_M/P \text { and }v_m=2\pi d_m/P\]
        
        :bulb:\[ r=a=d_m+d_M\]
    - - systems with more than a few stars
      - open clusters
        
        The Pleiades
        
        typically a few hundred stars
        
        irregular in shape
        
        hot, luminous stars
        
        close to plane of Galaxy
        
        associated with hot dust and gas clouds
        
        stars formed around the same time and region
        
        provide snapshot of same age/composition stars
        
        same distance from us
        
        allows us to compare relative properties
      - globular clusters
        
        Omega Centauri
        
        may contain thousands of stars
        
        tightly bound by gravity
        
        spherical shape
        
        little or no gas
        
        no hot stars
        
        probably very old
        
        studies of the history of Universe
        
        bright
        
        can be seen in other galaxies
        
        may help in judging distances to these galaxies
  - - - change as a result of a physical change in the star
    - - Change as a result of geometric effects
      - majority eclipsing binaries
    - - period of seconds to years
        
        regular variables #
        
        defined periods
        
        irregular variables
        
        some change continuously
        
        some suddenly change after a long period of not changing
        
        novae
        
        brightens up to 10 000 times in days
        
        10 magnitudes
        
        followed by a slow decline to normal luminosity
        
        not destroyed
        
        can be 1 observed outburst or many irregular outbursts
        
        days to years (irregular periods)
        
        supernovae
        
        one event
        
        star "dies"
        
        T Tauri stars
    - - 2/3 of variable stars
      - intrinsic variables
        
        luminosity, temp, and radius all changing
      - Cephids
        
        most important type of pulsating variable
        
        regular
        
        period of 1 day to 100 days
        
        luminosity changes up to a factor of 10
        
        important in studying life stages of stars
        
        provide a way to measure distance
        
        period-luminosity relationship
        
        absolute magnitude can be deduced from period and distance
        
        Henrietta Leavitt
    - - how magnitude changes with time
- - - - celestial sphere
        
        sphere of arbitrary radius around Earth
        
        projected flat onto star maps
        
        west is on right-side of star maps
        
        we're viewing the sphere from inside
      - limited
        
        don't show what part of sky visible at given time on Earth
      - celestial coordinates
        
        celestial longitude is right ascension
        
        measured in units of time
        
        Hours, minutes seconds
        
        zero point of RA is vernal equinox
        
        the celestial equator=the Earth's equator projected onto the celestial sphere
        
        celestial latitude is declination
        
        units of 0°-90°
        
        positive=north of celestial equator
        
        negative=south of celestial equator
  - - - easier to measure than true size
      - depends on object's distance
      - the Sun and Moon have same angular size
        
        0.5°
      - measured in degrees or radians
        
        degree subdivided into minutes of arc (arcminutes)
        
        divided into seconds of arc (arcseconds)
        
        radian little more than 57°
        
        1 radian=206265 arcsecs
- - - - can characterise size by radius
        
        most direct way to calculate radius is through angular diameter
        
        :bulb: \[R=[(\alpha/2)/\text {radians} ×d]\]
        
        where R is the star's radius, α is the angular diameter, and d is distnace
        
        provided α is small (always is)
- - - - Sun's "surface" temp
    - - comparing radiation emitted at two wavelengths/range of wavelengths
- - - - flux density
        
        measured in watts per square meter
        
        :bulb: given by\[F=L(4\pi d^{2})\]
        
        difficult to measure
        
        limited spectral range one detector will respond to
        
        requires multiple detectors
        
        e.g. X-ray detectors might not pick up visible light
        
        Earth's atmosphere
        
        not totally transparent at any wavelength
        
        opaque at many wavelengths
        
        most transparent at visible, some IR, most microwave and radio wave
        
        Solution: measurements at limited wavelength
        
        V-band/V-channel
        
        "visual"
        
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        all stars emit a large fraction of luminosity in V-band
        
        Earth's atmosphere transparent at these wavelengths
        
        \(\text {F}_V\) is flux density measured with a detector with spectral response of V-band
  - - - SI units measured in watts
      - determined by radius and temperature
        
        :bulb:\[L≈4 \pi R^{2}\sigma T^{4}\]
        
        σ is the Stefan-Boltzmann constant
        
        L is luminosity
        
        Ris radius
        
        T is average photospheric temp in K
        
        limited; number of stars with known radii in hundreds
        
        usually rearranged to find radius
      - standard candles
        
        luminosity of nearby object
        
        assume objects with similar properties have similar luminosities
      - Using V-band #
        
        :bulb: \[L_V=(4\pi d^{2})F_V\]
        
        \(L_V\) is measure of total output of a star in V-band
  - - - Hipparchus
        
        6 groups
        
        apparent visual brightness
        
        magnitudes
        
        each step is ~2.5 ratio in brightness
        
        logarithmic scale
        
        used and refined up until \(\text {F}_V \)could be detected
        
        brightest stars being first magnitude
        
        Apparent visual magnitude
        
        :bulb: \[(m_1-m_2)=-2.5 \log{(b_1/b_2)}\]
        
        b=apparent brightness
        
        can be F or other measurement (units cancel)
        
        m=apparent magnitude
        
        the flux densities are measured in V-band
        
        not intrinsic
        
        depends on distance
        
        Absolute visual magnitude
        
        intrinsic
        
        M=m-5 log d + 5
        
        :bulb: \[(M_{V1} -M_{V2})=-2.5 \log{(L_{V1}/L_{V2})}\]
        
        most stars range from -6 to 16 absolute visual magnitude
        
        Can be expressed in colours of stars
        
        UBV system
        
        colour index
        
        B-V
        
        measure of ratio of flux densities in B and V bands
- - - - b is the distance from the Earth to the Sun and d is the distance to the star
      - :bulb: Converting from parallax/arcsec to parsec: \(d/ \text {pc}=\frac {1}{p/\text {arcsec}}\)
        
        parsec is usually used to measure distances outside the Solar System
        
        1 ly =0.307 pc
      - Aberration of starlight must be accounted for.
        
        Direction of starlight result of speed of light and speed of Earth's orbit