Ch 07-01 Distance and Magnitude of Stars

• Distance determination & parallax
  
  
  
      d = 1/p
      p: parallax angle in arcseconds
      d: distance in parsecs (pc); this is how 'parsec' is defined.
   
  • Examples:
    Proxima Centauri, p= 0.772", d= 1.30 pc
    61 Cygni, p= 0.289", d= 3.46 pc
  • Ground-based telescopes can determine up to < 50 pc.
    The Hipparcos satellite, launched by ESA in 1989, can do it up to about 500 pc.
    (A star catalogue produced by the Hipparcos mission contains about two million stars.)
    The GAIA satellite, launched by ESA in 2013, can do it up to about 5 kpc to observe one billion stars.
  • Parallax is the cornerstone of all distance determination in astronomy.
  
   
• Proper motion of stars
  Besides parallax, stars do move in the sky. Nearby ones are easier to measure.
  
  Their motion in the sky represents their transverse speed:
  v_t = 4.74 μd
      v_t : transverse velocity in km/s
      μ: proper motion in arcsec/year
      d: distance in pc
  
  Their full speed can be obtained by incorporating the radial speed v_r:
  v=√(v_t²+v_r²)
  
  
   
• Magnitude
  
  • Apparent magnitude
    m - m₀ = -2.5 log(F/F₀)
      m: stellar magnitude
      F: observed flux
  • Absolute magnitude
    The apparent magnitude of a star if it were at a distance of 10 pc from the Earth
    F= L/(4πd²)
    m - M = -2.5 log[(L/d²) / (L/(10pc)²)]
    m - M = 5 log(d/pc) - 5
    e.g., for the Sun, m = -26.7, M = 4.8
     
    (m-M) is called the distance module
    If M can be somehow determined, with observed m, the distance is then
    d = 10^(m-M+5)/5 pc
    
    
     
• Color index
  
  
  
  Magnitudes in different bands
  
  m_x - m_x,0= -2.5 log[∫R_x(λ)f_λdλ / ∫R_x(λ)f_λ,0dλ]
  
  R_x(λ) is the instrument response (or sometimes called quantum efficiency), which is a function of wavelengths.
  
  
  
  * There are different conventions to define different bands. Some often used are U(3500Å), B(4400Å), V(5500Å), R(7000Å), I(9000Å), J, K...
  * The difference between magnitudes of two bands is called the color index. It actually indicates the surface temperature of a star. For example, the B-V color index:
      B-V = -0.4 corresponding to 50000 K
      B-V = 0.2 corresponding to 8000 K
  Our Sun has B-V = 0.656.