AH 1.6 Stellar Physics

Question 1

Define the apparent brightness of a star.

Answer 1

Question 2

The Luminosity of the Sun is 3.9 x 10²⁶ W m^-2. The Earth-Sun distance is 1.47 x 10¹¹ m.

Calculate the apparent brightness of the Sun.

Answer 2

b = L / 4πd²

= 3.9 × 10²⁶ / [4π x (1.47 x 10¹¹ )²]

b = 1.44 x 10³ Wm^-2

Question 3

What knowledge of a star is required in order to calculate it’s surface temperature?

Answer 3

The colour of the star or, more specifically, it’s spectrum.

Question 4

The surface temperature of the sun is 5800 K. Find the power it emits per unit area.

The Stefan–Boltzman constant = 5.67 × 10^–8 W m^–2 K^–4

Answer 4

P = σT⁴

= 5.67 × 10^–8 × (5800)⁴

P = 6.42 × 10⁷ Wm^–2

Question 5

Derive the equation 𝐿 = 4𝜋𝑟²𝜎𝑇⁴ …taking care to explain what each symbol means

Answer 5

For any star radius r, the Power emitted per unit area is given by

P= σT⁴ …σ = the Stefan-Boltzman constant & T = surface Temperaure

The Luminosity L of a star is the total rate at which energy is radiated by its whole surface over all wavelengths. So

L = σT⁴ x 4πr² … as the srface area of a sphere is 4πr²

Question 6

Explain the concept gravitational equilibrium in a star.

Answer 6

Once a star is on the main sequence it is in state of hydrostatic equilibrium.

This means that the inward force of gravitational attraction between all of its own molecules is balanced by the outward force of thermal pressure caused by the heat.

These two forces balance each other, ensuring that the star’s radius remains constant.

Whilst this hydrstatic equilibrium is maintained, the star remains on the main sequence of the Hertzsprung-Russell diagram.

Note

In general there is a balance between Gas Pressure & Gravity.

If Pressure dominates, the star expands.

If Gravity dominates, the star contracts.

I_f Pressure and Gravity balance then the start is in hydrostatc equilibrium and has constant radius._

Question 7

Describe the fusion process that takes place in stars known as the proton-proton chain

Answer 7

Nuclear fusion in the Sun converts Hydrogen into Helium with the release of energy.

The process is summmarised below:

4 (1H) → 4He + 2e⁺ + 2 neutrinos + energy

However, the process occurs in several steps, shown in the diagram.

Question 8

Describe the formation of stars.

Answer 8

Molecular clouds called nebulae form mostly from hydrogen, with lesser quantities of other molecules such as carbon monoxide, ammonia, water and ethanol. In addition, small quantities of dust are commonly found, consisting of carbon, iron, oxygen and silicon.

The molecules are attracted to each other gravitationally. In nebulae, the gravitational attraction between the molecules overcomes the thermal pressure due to their motion and progressively the molecules ‘clump’ together and gradually compress into a core. The process steadily heats the core (the particles move faster) until it reaches a temperature where nuclear fusion can be sustained and radiation is emitted - a star has been formed.

Question 9

The Luminosity and surface temperature of a star are determined by one physical quanity. What is it?

Answer 9

The mass of the star.

(This, of course, depends on the initial conditions within the nebula in which the star was formed)

Question 10

Explain what is meant by the Hertzsprung-Russell diagram and comment on its axes.

Answer 10

The Hertzsprung-Russell diagram is a graphical plot of stellar luminosity against temperature for many stars relatively close to Earth.

On the x-axis, temperature decreases to the right. The scale is logarithmic.

On the y-axis, luminosity increases going up. The scale is logarithmic

Question 11

Sketch a Hertzsrung-Russell diagram and on it clearly draw and label

a) the main sequence
b) red giants
c) supergiants
d) white dwarves

Answer 11

Question 12

What process is going in and so characterises a main sequence star?

Answer 12

The nuclear fusion of Hydrogen into Helium.

Question 13

What two important properties of stars change over time?

Answer 13

The luminosity and temperature of stars change over time.

Question 14

Which stars have a longer lifetime - high or low mass stars?

Answer 14

Low mass stars have a longer lifetime.

Question 15

Why do higher-mass stars have a shorter lifetime than lower-mass stars?

Answer 15

Higher mass stars have a shorter lifetime since they use up their mass more

quickly.

Question 16

Describe in detail the final stages in the life of a star after it leaves the main sequence.

Answer 16

Low-mass stars (< 8m_Sun)

1. The star eventually begins to run out of hydroegn fuel. This causes the stellar pressure to decrease so allowing gravitational pressure to compress the core. This causes the temperature to rise and so the star’s luminosity and volume increase. The increase in volume means that the surface temperature cools. The star passes through the subgiant region to become a red giant.

2. As the red giant continues to expand it reaches such a large volume that its outer layers cannot be contained by gravitational attraction. The outer layers gradually disperse to form a so-called planetary* nebula and only the dead stellar core is left. No fusion can now occur so the core cools. The star has become an inert white dwarf (consisting mostly of hydrogen and oxygen).

Higher-mass stars (> 8m_Sun)

1. In the early stages the evolution of a higher-mass star is similar to that of lower mass stars.

2. However, when the Hydrogen runs out the fusion of Helium into Carbon begins, followed by further nucleosynthesis producing heavier elements, ending with iron. The expansion is also much greater and the star becomes a red supergiant.

3. The iron pruduced by nucleosythesis accumulates in the core rendering the mass and density of the star so high that the core collapses in on istelf under its own gravity. Such is the gravitational pressure that protons and electrons are forced together to form neutrons. The density is now so great that the diameter of the star is now only a few km and is known as a neutron star.

Very high-mass stars (> 40 m_Sun)

If the original stellar mass is very large (> 40m_Sun) then the gravitational collapse is so great that a black hole is formed, rather than a neutron star.

* Note that this has nothing to do with planetary formation - it is just an historical term.

Question 17

The ultimate fate of a star e.g. whether it ends up as a white dwarf, a neutron star or a black hole, depends on what physical property of the star?

Answer 17

The mass of the star.