Measuring Stellar Distances

Venus, Jupiter and the Moon 17/11/2017

Three useful units – which we use depends on how far away the object is. Being able to convert quickly is useful to know.

• Relatively small distances – Astronomical Unit, AU = average distance between Earth and Sun  (1 AU=150 million km)
• Light Year. 1 ly is the distance light travels in a year at a speed of 300 million m/s.

Find the distance in km between us and our nearest star Proxima Centauri (4.3ly away) You might like to speculate how long it might take a spacecraft travelling at a maximum possible 25,000km/h to reach it.

• One parsec (pc) is defined as the distance to a star that shifts by one arcsecond from one side of Earth’s orbit to the other. These angles are incredibly small. They’re too small for degrees to be a practical unit of measurement. There are 3,600 arcseconds (60 minutes x 60 seconds) in one degree. To provide some perspective: one arcsecond is equivalent to the width of an average human hair seen from 20m away.

The nearest star is Proxima Centauri, at 1.3 pc. The Andromeda Galaxy, the closest spiral galaxy to our own, is nearly 800 kiloparsecs away.

If we imagine ourselves taking measurements of an imaginary star at six monthly intervals, it seems to have moved with respect to the Sun, 1 AU away.

Putting this another way:

Explaining Trigonometric Parallax.

• 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 second 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. Shut one eye and hold your finger about eighteen inches in front of your face. Observe a distant object and the finger. Keeping still, look with the other eye. The finger represents the near star and appears to have moved with respect to the background. If you ask a friend to hold up his finger and repeat the observation, it would seem to have moved much less.
• The parallax or apparent shift (from the Greek for ‘alteration’) 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 as shown. If we measure the distance to the star in AU. (astronomical units), then the parallax is given by:

For example – the six month parallax angle for Alpha Centauri is 1.52 seconds of arc. You might like to calculate how far away this is in light years.

Luminosity

A supernova

• Is the amount of electromagnetic energy a body radiates per unit of time. {J/s (W)}
• Is intrinsic to a body and is a measurable property which is independent of distance.

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 brightness.

A is the area of the illuminated surface, a sphere of radius r.

F is the Flux of the illuminated surface, the energy radiated per second per square metre of surface from a point source such as a star. This means that Luminosity is the total flux in watts

So:

The sun has a luminosity L = 3.8 x 10²⁶W. If Earth – sun distance is 150m km and the Sun can be considered a point source, we can show that the radiant energy flux at the surface of the Earth is about 1.3kW m^-2

Hertsprung and Russell showed that that the luminosity of a star L (assuming the star is a black body – a perfect emitter and absorber – which is a good approximation) is also related to temperature T and radius R of the star by the equation:

where  ‘sigma’ is the Stefan-Boltzmann Constant  5.67 × 10⁻⁸ W·m^-2·K^-4

L is often quoted in terms of solar luminosities, or how many times as much energy the object radiates than the Sun, so L_Sun= 1

Questions:

1. Find the actual luminosity of the Sun, given a surface temperature of 6,000K and a radius of 7 x 10⁸m
2. Compare with Sirius – a very bright star – temperature 12,000K and radius 2.22 x 10⁹m. How much brighter is Sirius than the Sun – in solar luminosities?
3. What would be the surface temperature of a star having the same luminosity as the Sun but twice the radius. What would it look like?

• So a bigger star can be at a lower temperature and yet have the same luminosity, i.e. it looks just as bright
• A hotter star is more luminous than a cooler one of the same radius.
• 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.

Finally, the idea of a Standard Candle is an important one in astrophysics. Certain classes of objects such as supernovae and Cepheid variable stars have properties whereby their luminosities can be determined separately from other measurements. For example, the period of a Cepheid variable star depends on it’s mean absolute magnitude; the more luminous the star, the longer the period.

If we know the luminosity and can measure the energy flux or brightness, then by comparing it with a standard candle then we can figure out how far away it is by comparing it with a standard candle of the same luminosity.