**What we now know** lmax T = 2.8977 x 10-3 • Temperature of stars • Use stellar spectra – find lmax – use Wien’s law to find the temperature • The compositions of stars: 75% H, 24% He and 1% every thing else • Spectral sequence: OBAFGKM – a method of finding similar temperature stars • Relationship between Flux, Luminosity and distance

**Stellar luminosity** If we combine the flux measurement F and the distance measurement d for a star, then we can find its luminosity Why! You measure its parallax me old befuddled chum So, how do we find the distance to a star? A fundamental star property or,

**Distances from Angles** The key to finding distance is to measure position angles from two separate locations 2 a X D 1 With X and a being measured river Elementary my dear Watson

**Distances to the stars** Direct method: stellar parallax no assumptions need be made about the star Observations: measure the apparent shift in the position of a near-by star over a 6 month time interval The point! The Earth moves half way around its orbit in this time interval

**Stellar Parallax – an apparent shift in the position of a** star due to Earth’s motion about the Sun During a 6 month time interval the nearby star moves through a measurable angle WRT the much more distant background stars

**And the geometry says:** Distant stars Obs. 2 Earth’s orbit Distance, d 1 AU P Sun Near-by star Obs. 1 Idea is that observations give P the angle of parallax

**Observations reveal the 6-month motion of the near-by star** Time from Obs. 1 to Obs. 2 = 6 months View from Obs. 1 View from Obs. 2 Measure: Angular shift = 2 P Note: the further a star is away, so the smaller is the angle of parallax

**Parallax motion in the direction of Orion**

**Trigonometry: Tan(P) = 1 AU / distance (d)** The clever bit: for small angles Tan(P) P(radians) P(radians) = (2 p / 360) P(degrees) P(arcsec.) = P(degrees) x 3600 Result: d = 206,265 AU / P(arcsec.) WTF See triangle handout purely from numerical conversion terms

**Notes & definition:** if P = 1 second of arc, then d = 206,265 AU Hence: if the observed angle of parallax is 1 arc second the distance is defined to be 1 parsec so the conversion factor is 1 parsec = 206,265 AU Final result: d(parsec) = 1 / P(arcsec.) OOTETK

**Word origins** Parsec The concatenation of parallax and arc second 1 parsec is the distance to an object with a parallax of 1 arc second And doing the Kessel Run in under 12 parsecs just doesn’t make sense – boast you too much me think Han Solo – and fail you Astronomy 101

**Proxima Centauri** Robert Innis – discovered Proxima in 1915 Nearest star to the Solar System and Sun Observed parallax shift on the sky over a six month time interval is Angle = 1.5377 arc seconds = 0.00042714 degrees = 2 x P Proper motion over 25 years Angle of parallax = P = 1.5377 / 2 = 0.76885 arc seconds Distance to Proxima = 1 / P = 1/0.76885 = 1.301 parsecs

**Let’s get Sirius** Distance to Sirius = 8.6 light years What is its angular shift on the sky over a six month interval ?

**Given: d(Sirius) = 8.6 light years = 2.64 pc** So, from Parallax formula (after algebra switch) And hence 6 month angular shift = 2 x P = 0.758 arcsec (equivalent to 2 x 10-4 degrees = human hair at 28-m) Unit conversion Oh! To be far from Azkaban and doing astronomy 101

**Parallax ain’t easy** Parallax shifts are fractions of an arc second ground based limits: P > 0.05 arcsec. (d < 20 pc) space based limits: P > 0.002 arcsec. (d < 500 pc) Beyond about 500 pc must use indirect methods Find and calibrate “standard candles” Definition:Objects of known luminosity (e.g. sun-like stars)

**First parallax numbers (1837)** No need to copy table (Sun-like star)

**All systems are go….** Calibration step: Stars with measured flux and distance d and spectral type Calculate temperature and luminosity Calibrate spectral type with luminosity = standard candle Once parallax too small to measure Find distance from spectral type - luminosity calibration We can now find the distance to any star in the observable universe – cool from spectral type calibration measured by parallax