Distances of the Stars

Key Ideas• Distance is the most important & most difficult

quantity to measure in Astronomy

• Method of Trigonometric Parallaxes– Direct geometric method of finding distances

• Units of Cosmic Distance:– Light Year – Parsec (Parallax second)

Why are Distances Important?

• Distances are necessary for estimating:– Total energy emitted by an object (Luminosity)– Masses of objects from their orbital motions

• True motions through space of stars• Physical sizes of objects

The problem is that distances are very hard to measure...

The problem of measuring distances

• Question: How do you measure the distance of something that is beyond the reach of your measuring instruments? – Examples of such problems:

• Large-scale surveying & mapping problems.• Military range finding to targets• Measuring distances to any astronomical object

• Answer:– Radar - Cepheids -

Redshift &– Parallax - Supernovae

Hubble’s law

Radar

• We know the speed of light: 3E8 m/s

• Measure time for light to go to an object and return– Calculate distance,

d=vt– Works for distances

within solar system

Measuring Parallax• Go to one corner of the room• Use the compass on your phone to measure the azimuth

(angular direction) to one of the hanging objects• Go to the other corner & measure the azimuth to the same

object again.• Find the distance between the two corners• Calculate the distance to the object

b = baselined = distance to objectα = measured angle

b

dα

b/2 ÷ tan α = d

The Method of Trigonometric Parallaxes• Nearby stars appear to move with respect to more distant

background stars due to the motion of the Earth around the Sun. • This apparent motion (it is not "true" motion) is called Stellar Parallax. • In the picture, the line of sight to the star in December is different

than that in June, when the Earth is on the other side of its orbit. • As seen from the Earth, the nearby star appears to sweep through the

angle shown. • Half of this angle, is the parallax, p.

b = baselined = distance to objectα = measured parallax angle

b

d

α/2

α

Parallax decreases with Distance

• In the upper figure, the star is about 2.5 times nearer than the star in the lower figure, and has a parallax angle which is 2.5 times larger.

• This gives us a means to measure distances directly by measuring the parallaxes of nearby stars.

• We call this powerful direct distance technique the Method of Trigonometric Parallaxes.

Stellar Parallaxes• The nearest stars are very far away

– the largest measured parallaxes is very small; less than an arcsecond.

– For example, the nearest star, Proxima Centauri, has a parallax of 0.772-arcsec (the largest parallax observed for any star).

• First parallax observed 1837 (Friedrich Bessel) for the star 61 Cygni.

• Today measuring parallaxes:– Use of photography and digital imaging techniques– Satellite measurements avoid blur due to the Earth's

atmosphere.

Parallax Formula• We saw before that the smaller the parallax, the

larger the distance. We can express this as a simple formula:

• Where: p = parallax angle in arcsecondsd = distance in "Parsecs"

• Writing our parallax formula in this way allows us to define a new "natural" unit for distances in astronomy: the Parallax-Second or Parsec.

Limitations

• If the stars are too far away, the parallax can be too small to measure accurately.

– The greater the distance, the smaller the parallax, and so the less precise the distance measurement will be.

– The smallest parallax measurable from the ground is about 0.01-arcsec. This means that from the ground, the method of Trigonometric Parallaxes has the following limitations.

• good out to 100 pc• only get 10% distances out to a few parsecs• only a few hundred stars are this close

Solutions: Satellites• Hipparcos

• Hipparcos measured parallaxes for about 100,000 stars• Got 10% accuracy distances out to about 100 pc• Good distances for bright stars out to 1000 pc.

• Space Interferometry Mission (SIM)• Parallax precision of ~4 microacseconds• 10% distances out to 25,000 parsecs, • encompassing the Galatic Center (8000pc away) and the halo of the

Galaxy.

• Gaia• Measure positions and motions for about 1 Billion Stars• Measure parallaxes for >200 Million Stars• Max precision of 10 microarcseconds• This means 10% or better distances out to 10,000pc

Parallax Second = Parsec (pc)

• Fundamental unit of distance in Astronomy– "A star with a parallax of 1 arcsecond has a

distance of 1 Parsec." • 1 parsec (pc) is equivalent to:

– 206,265 AU – 3.26 Light Years – 3.086 x 1013 km

Light Years

• An alternative unit of astronomical distance is the Light Year (ly).– "1 Light Year is the

distance traveled by light in one year."

• 1 light year (ly) is equivalent to:– 0.31 pc – 63,270 AU

• Used primarily by writers of popular science books and science fiction writers. It is rarely used in research astronomy.– The parsec is directly derived

from the quantity that is being measured (the stellar parallax angle)

– The light-year must be derived from having previously measured the distance in parsecs.

– In this way, the parsec is a more "natural" unit to use than the light year.

Measuring the Universe

<iframe src="//player.vimeo.com/video/41434123?autoplay=1" width="500" height="375" frameborder="0" webkitallowfullscreen mozallowfullscreen allowfullscreen></iframe> <p><a href="http://vimeo.com/41434123">Measuring the Universe</a> from <a href="http://vimeo.com/royalobservatory">Royal Observatory Greenwich</a> on <a href="https://vimeo.com">Vimeo</a>.</p>

Magnitude (History)• Magnitude is a measure of the brightness of a

star.• Hipparchus, 150 BC, developed a scale

1 = brightest stars & 6 = dimmest starsDid not include the sun, moon, or planets

• Norman Pogson, 1856– expanded Hipparchus’ scale to include things we

can see with a telescope (Even HST)– Negative values - brightest (Sun, moon, planets)– He noted that we receive 100 times more light

from a first magnitude star as from a sixth.

Apparent Magnitude Celestial Object

-26.7 Sun

-12.6 Full Moon

-4.4 Venus (at brightest)

-3.0 Mars (at brightest)

-1.6 Sirius (brightest star)

+3.0 Naked eye limit in an urban neighborhood

+5.5 Uranus (at brightest)

+6.0 Naked eye limit

+9.5 Faintest objects visible with binoculars

+13.7 Pluto (at brightest)

30 Faintest objects observable by the Hubble Space Telescope

Orders of Magnitude• Brightest stars = first order of magnitude (magnitude = 1)• dimmer stars were 2nd, 3rd, 4th order, etc. (magnitudes = 2, 3, 4…)• (m2-m1) = 2.5 log10 (b1/b2)

Apparent Magnitude Difference (m2-m1)

Ratio of Apparent Brightness (b1/b2)

1 2.512

2 (2.512)2 = 6.31

3 (2.512)3 = 15.85

4 (2.512)4 = 39.82

5 (2.512)5 = 100

10 (2.512)10 = 10,000

20 (2.512)20 = 100,000,000

Magnitude (Brightness)

• Negative numbers are the brightest, positive numbers are the dimmest.– The naked eye can see down to around the sixth

magnitude (that is +6).– Galileo saw about magnitude 9 with his telescope. – The Hubble Space Telescope? About magnitude

29.

Magnitude (Apparent vs. Absolute)

• Apparent Magnitude: How bright a star appears from Earth

• Absolute Magnitude: How bright a star really is (if you could see them all from the same distance)

Magnitude (Apparent vs. Absolute)

Examples: • The star Betelgeuse has an apparent magnitude

of 0.5 because it is so far away, but if you could go closer to it, you would see that it is VERY bright and has an absolute magnitude of -7.2

• The sun is so close to us that it has an apparent magnitude of -26.7, but if you compare it to other stars, it isn’t really very bright and has an absolute magnitude of only 4.8

Magnitude and Color• In practice, magnitude is measured in wavelengths informs

about the surface temperature.• Surface temperature determines color.

– Blue stars hotter than yellow stars hotter than red stars. • Sirius is HOT, 9,400 K. Emits more blue than red light• Betelgeuse is COOL, 3,400 K. More red than blue

– Color Index of a star• the difference between the magnitude of the star in one filter and the

magnitude of the same star in another filter.• Any filters can be used for color indices, but some of the most common

are B - V (Blue – Green) and V – R (Green – Red). • Remember that magnitudes decrease with increasing brightness, so if B -

V is small, the star is bluer (and hotter) than if B - V is large.

• Example: – B = 6.7 and V = 8.2: B - V = 6.7 – 8.2 = -1.5– B = 6.7 and V = 5.8: B - V = 6.7 - 5.8 = 0.9

Why aren’t there any green stars?

Cepheids• AKA: Cepheid Variables

– Stars brighten & dim very predictably

• 1912 – Henrietta Swan Leavitt– Measured time & brightness of 25 Cepheids– Period of pulses proportional to Luminosity/brightness

Cepheids• Brightness is inversely

proportional to distance• We can determine actual

magnitude!• Difference in actual

magnitude (M) vs. apparent magnitude (m) gives distance:

m-M = -5 + 5 log (d)

Type Ia Supernovae

• Gravity increases to 1.44 solar masses– Inner layers collapse– Outer layers explode

• Exploding White Dwarfs with companion stars• No H-lines in Spectra• Gravity of White Dwarf allows it to take matter from its

companion

Using Supernovae to find Distance• All Type Ia reach same

brightness– M=-19.3 + 0.03

• Distinct dimming curve• Comparing apparent

brightness to known actual brightness:

m-M = -5 + 5 log (d)

Red Shift & Hubble

• Distances > 1 Billion LY• Expanding Universe

– All objects moving away from each other

v = Hdv = outward velocityd = distance from EarthH = Hubble Constant ≅ 65 km/sec/Mpc

• Doppler Red Shift (v)