# Activity: Longer distances and mass

#### Aims

• Introduce spectral classification.
• Understand how the HR diagram can be used to determine distance
• Introduce the concept of standard candle.
• Understand why Cepheid variables have varying luminosity.
• Understand the use of Cepheid variables to measure distance.
• Use Kepler's law to determine mass.
• Use the mass luminosity relationship to determine mass.

#### Spectral classification

This is not on the syllabus but will help you to understand information from stellar databases

From the intensity of the Hydrogen absorption lines in a stars spectrum it is possible to determine the temperature of a star. The star is then allocated a letter to represent its temperature. This is known as Harvard classification.

ClassTemp/KColour
O (Oh)60,000blue
B (Be)30,000blue-white
A (A)10,000white
F (Fine)7,500yellow - white
G (Girl)6,000yellow
K (Kiss)5,000orange
M (Me)3,500red
• What is the spectral classification of Betelgeuse?

#### Luminosity classification

By observing the absorption lines closely we see that the larger stars have narrower lines.

• One of the following spectra is from a main sequence star the other from a supergiant, which is which? In this way stars of the same temperature can be ordered according to size, this is denoted by a roman numeral from I to VII known as the Yerkes scale.

Luminosity classstar type
Isupergiant
IIbright giant
IIIgiant
IVsubgiant
Vmain sequence
VIsubdwarf
VIIdwarf

#### Spectral parallax

This is the way a stars spectrum can be used to determine distance (nothing to do with parallax by the way). Once the spectral and luminosity classes have been determined the star can be put onto the HR diagram. This method can be used for stars that are too far for the parallax method to be used.

Sirius has a spectral classification A1V (the 1 is a sub division of A)

• By placing Sirius on the HR diagram determine its luminosity. Note that this HR diagram has the Harvard class on the x axis.

• Given that the apparent brightness of Sirius is 10-7 Wm-2 show that it is 8.6 ly from the Earth

#### Cepheid Variable

If we know the luminosity of a certain type of star then by measuring its brightness then determine its distance. Such a star is called a standard candle. A candle flame is always about the same size so if you look at candles you can know which are further away.

Christmas tree lights can also be used. Here you can see which lights are closest.

When a star explodes into a Super Nova it always does so with the same luminosity so these events can be used. This doesn't happen so often but it is also possible to determine the luminosity of a variable star such as a Cepheid variable.

A Cepheid is a star whose brightness varies due to its changing size.

1. The core of the star is producing heat which is absorbed by the outer layer causing the outer layer to get hot.
2. As the outer layer gets hot it expands and cools.
3. The outer layer now absorbs less radiation. This means that more radiation is escaping which results in an increase in brightness.
4. As radiation escapes the gas cools. The cooler gas now contracts due to the force of gravity. This causes it to get hot so it absorbs more radiation and the cycle repeats. The special thing about Cepheids is that the time period of their change in brightness is related to their luminosity in such a way that a log - log graph of L vs T will give a straight line. Using this line the luminosity of distant Cepheids can be found. Note that there are many other variable stars but Cepheids can be identified by the characteristics of their light curve. To find variable stars photographs are used in a clever way. You can try this on on the University of Nebraska - lincoln's blink simulation. • The observation list shows the same stars at different times.
• Add a series of images.
• Can you spot the difference as you add the images?
• Do you see it now?

#### Measuring mass

Without binary stars measuring the mass of a star would be rather difficult, luckily almost half of the stars we see are actually two stars orbiting each other. Some of these can actually be seen orbiting when viewed through a telescope others are too far away but we can tell that there are two stars as they eclipse each other.

• You can see how this looks using the binary star simulation from University of Nebraska - lincoln.

• Run the animation and see how the brightness varies as the stars eclipse each other.

From the light curve we can find the period of the orbit. if we consider stars of equal mass with the same orbital radius we can see that the heavier stars have a shorter time period. This is illustrated in the simulation here.  According to Keplers Laws if the radius is the same the total mass (M) is inversely proportional to the time period squared. • By measuring the time periods determine the ratio of masses.

If the masses aren't the same then the orbits will not be identical but you can use the ration of orbit radius to find the ratio of mass.

• Can you tell which of the following situations has the biggest difference between the big mass and the small one.  #### Mass Luminosity relationship

By measuring the mass a luminosity of many main sequence stars it was found that. So once you know L you can find M

• Show that the mass of Sirius is 2 Ms .
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