Astronomical distances. https://pixabay.com/static/uploads/photo/2014/11/09/11/08/measuring-tape-523525_960_720.jpg Astronomical distances. Option D: Lesson 5

Astronomical distances The SI unit for length, the metre, is a very small unit to measure astronomical distances since the distances between astronomical objects are really really large Here are some units usually used is astronomy: The Astronomical Unit (AU) – this is the average distance between the Earth and the Sun. This unit is more used within the Solar System. 1 AU = 150 000 000 km or 1 AU = 1.5x1011m

Astronomical distances The light year (ly) – this is the distance travelled by the light in one year. c = 3x108 m/s t = 1 year = 365.25 x 24 x 60 x 60= 3.16 x 107 s Speed =Distance / Time Distance = Speed x Time = 3x108 x 3.16 x 107 = 9.46 x 1015 m 1 ly = 9.46x1015 m

Astronomical distances The parsec (pc) – this is the distance at which 1 AU subtends an angle of 1 arc second. “Parsec” is short for parallax arcsecond 1 pc = 3.086x1016 m or 1 pc = 3.26 ly

Mathematical Aside… A minute of arc, arcminute, or minute arc (MOA), is a unit of angular measurement equal to one sixtieth (1⁄60) of one degree (circle⁄21,600), or (π⁄10,800) radians. In turn, a second of arc or arcsecond is one sixtieth (1⁄60) of one minute of arc. (thanks wikipedia…) So. 1 arc second is 1/60th of 1/60th of a degree, or 1/3600th of a degree. 1 arc second = 0.00278 degrees or 1 arc second = 4.8x10-8 radians. That’s a small angle yo!

Angular sizes 360 degrees (360o) in a circle 60 arc minutes (60’) in a degree 60 arc seconds (60”) in an arcminute (so 3600 arc seconds in a degree!)

1 parsec = 3.086 X 1016 metres Nearest Star 1.3 pc (206,000 times further than the Earth is from the Sun)

Measuring Distances Ok, but how do we know distances to the stars? There are few different methods. Parallax. Spectroscopic Parallax Using a “standard candle”.

Parallax. Parallax uses the motion of the Earth around the sun to determine the distance to distant stars. This is very similar to how our brains use two eyes to estimate distance to objects. Ever noticed how if you are driving in a car, objects that are really far away (like the moon or mountains?) don’t really seem to move relative to you, but objects that are close (like trees or lamp posts) seem to move? Hold a finger out in front of your face, and look at it through one eye (with the other eye closed) comparing it’s location to a distant object (picture on the wall or something out the window). Now close that eye and open the other. See how your finger seems to move? Parallax uses this to measure the distance to stars. It works b/c the earth orbits the sun, allowing us to make measurements of angles to objects from two different locations.

Parallax Parallax (more accurately motion parallax) is the change of angular position of two observations of a single object relative to each other as seen by an observer, caused by the motion of the observer. Simply put, it is the apparent shift of an object against the background that is caused by a change in the observer's position.

Parallax If you look at the diagram, it’s just trigonometry. We figure out the Parallax angle (P) by measuring the angle from the Earth (relative to really distant objects) to whatever object we wish to know the distance to. We do this at two two different times of year exactly 6 months apart. To determine distance, we use the Tangent function. The “Opposite” leg of the triangle (R) is the distance between Earth and Sun(1 Au). The distance (d) to the object is the “Adjacent”.

Parallax For very small angles (using radians!!!) we approximate tan p ≈ p so: or:

Parallax The definition of parsec was the distance subtended by 1 arc second, and 1 arc second (in radians) is (2π/360)x(1/3600) and so

Parallax has its limits The farther away an object gets, the smaller its apparent shift. Eventually, the apparent shift in the star’s position due to motion of the Earth too small to Measure

Limits of Parallax Parallax only works for stars closer than about 100 parsecs. When objects are at distances greater than 100 parsecs, the change in angle due to motion of the Earth is too small to measure, and parallax measurements are not useful.

Spectroscopic Parallax and Standard Candles Both these methods rely on the fact that energy that we receive from a star every second (“apparent brightness”) depends on the distance we are from the star, and the star’s luminosity. Where b is apparent brightness, the energy we measure at Earth each second. L is luminosity of the star. d is distance to the star. (This is because the Luminosity is energy sent out in every direction. This energy gets spread out over a surface area the size of a sphere with radius “d”). So if we know the Luminosity of a star, and we measure its brightness. We can determine distance.

Spectroscopic Parallax Spectroscopic Parallax is not actually “Parallax” (ie it doesn’t use trig, or the shift in position of the star). Spectroscopic Parallax uses the luminosity and apparent brightness of a star. To determine the Luminosity of a star, we look at its: absorption spectra (to figure out spectral class). the peak wavelength of the radiation received from the star to determine it’s surface temperature (Wein’s Law). We then assume that the star is a main sequence star, and use the HR diagram to estimate the star’s luminosity. Once the Luminosity is known and the brightness we can determine distance. http://www.astro.cornell.edu/academics/courses/astro201/hr_diagram.htm For this to work, stars must be Main Sequence stars Within 10 MegaParsec.

Cepheid Variables and The Standard Candle Once stars are more than 10 Mpc, its hard to distinguish the difference between magnitude of stars and their distance. To determine distances to distant galaxies, we use Cepheid Variables. Since the period of “pulsing” of Cepheid Variables depends on the Luminosity of the stars. We can determine their Luminosity by measuring the amount of time it takes them to cycle through one period. Once the luminosity of a star is known, we can use the apparent brightness to determine the distance. We can then determine distances to the galaxies that contain these Cepheid Variables. Since CVs are of Known Luminosity in a particular galaxy, we can compare the other stars in that galaxy to those stars. So the CVs act like a “standard candle” (a star of known luminosity, a measuring stick of sorts).

Standard Candles We can also use the fact that Type Ia Supernovae have very predictable Luminosities. If we observe one of these supernovae, we can determine the distance to the galaxy in which they are in. We can also use binary stars as “standard candles”. If we observe Binaries, we can determine their period and therefore their mass. Once we know their mass, we can determine their luminosity. Once we know their luminosity we can determine distance.

Cepheid Variables AGAIN! Cepheid Variables are those whose absolute Magnitude (or luminosity) varies periodically The period of variation is related to their absolute magnitude (or luminosity) Distance measurement method Measure apparent brightness of the star (b) Measure period (T) Use period-luminosity law to find L Use the equation below and find distance Note from the graph below, the relationship between magnitude (M) and period of the Cepheids…

Distance measurement – Spectroscopic parallax (review) method (up to 10 Mpc) Step1 – Observe the star’s spectrum (with instruments) and identify its spectral type Step2 – Get the luminosity (L) of the star from the HR diagram Step3 – Measure (with instruments) the star’s apparent brightness (b) Step4 – Calculate the distance using the formula