Download Distance and Luminosity (new 2012)

How Bright and how Far Away…
Measuring Distances.
Light Year, ly. 1 ly is the distance light travels in a year. Find the distance in km between us and
Proxima Centauri (4.3ly away)
Astronomical Unit, AU = average distance between Earth and Sun (1 AU=150 million km)
Parsec, pc
A parsec is the distance at which a star would be if its parallax were exactly 1 second of arc. The
nearest star is Proxima Centauri, at 1.3 pc
Convert 1pc into
I.
AU
II.
ly
III.
km.
Trigonometric parallax
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Because of the Earth's revolution around the Sun, nearby stars appear to move with respect
to very distant stars which seem to be standing still.
Measure the angle to the star and observe how it changes as the position of the earth
changes. In the diagram if the observation point is at the top of the picture, six months later
it will be at the bottom, 2 AU’s away
You can use your fingers to show trigonometric parallax
The parallax of a star is the apparent angular size of the ellipse that a nearby star appears to
trace against the background stars. Because all parallaxes are small (the stars are very far
away), we can use the small angle approximation
tan 𝜃 ~𝜃𝑟𝑎𝑑 , for small 𝜃. If we measure the distance to the star in A.U. (astronomical units),
then the parallax is given by:
𝜃𝑟𝑎𝑑 =
1
𝑑
Luminosity L
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Is the amount of electromagnetic energy a body radiates per unit of time. {J/s (W)}
Is an intrinsic measurable property independent of distance.
depends on temperature and surface area of the star
Imagine a point source of light of luminosity L that radiates equally in all directions. A hollow sphere
centred on the point would have its entire interior surface illuminated. As the radius increases, the
surface area will also increase, and the constant luminosity has more surface area to illuminate,
leading to a decrease in observed or apparent brightness, b.
𝑏=
Where:
𝐿
𝐴
A is the area of the illuminated surface.
b is the apparent brightness or emitted power per unit area of the illuminated surface. 𝑊𝑚−2
For stars and other point sources of light, 𝐴 = 4𝜋𝑟 2 so
𝑏=
𝐿
4𝜋𝑟 2
Where:
𝑟 is the distance from the observer to the light source.
Verify that the answer should be 1300Wm-2
Calculate your answer in m and convert to pc. Ans = 200pc
If we use a detector, the received energy per unit detector area per second falling on the detector
can be written:
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑑𝑒𝑡𝑒𝑐𝑡𝑜𝑟
𝑎𝐿
𝑟𝑒𝑐𝑒𝑖𝑣𝑒𝑑 𝑒𝑛𝑒𝑟𝑔𝑦 /𝑠 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑝ℎ𝑒𝑟𝑒 𝑥 𝑙𝑢𝑚𝑖𝑛𝑜𝑠𝑖𝑡𝑦 = 4𝜋𝑟2
Hertsprung and Russell showed that that the luminosity of a star L (assuming the star is a black
body1, which is a good approximation) is also related to temperature T and radius r of the star by the
equation:
𝐿 = 4𝜋𝑟 2 𝜎𝑇 4 = 𝐴𝜎𝑇 4
where 𝜎 is the Stefan-Boltzmann Constant 5.67 × 10−8 W·m-2·K-4
1
Perfect absorber, perfect emitter of radiation – a perfect radiator
L is often quoted in terms of solar luminosities, or how many times as much energy the object
radiates than the Sun, so LSun=1
Compare with this…
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So a bigger star can be at a lower temperature and yet have the same luminosity, i.e. it
looks as bright

A hotter star is more luminous than a cooler one of the same radius.
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A bigger star is more luminous than a smaller one of the same temperature.
A cool (red) giant star is more luminous than the Sun because, even though it is cooler, it is much
larger than the Sun.
Questions.
a. Radius of star A is three times that of B, and its temperature is twice as high. Find the
𝐿
fraction 𝐿𝐴 .
𝐵
b. If they have the same apparent brightness, what is the ratio of their distances
c. A star has twice the sun's surface temperature and 1000 times its luminosity. How
many times bigger is it?
d. A star's apparent brightness is 16nWm-2 at a distance of 32ly. Find its luminosity.