Stars and astrophysics

IB Physics Higher Level· Theme E.3: Stars· 25 min read

1. Stellar Formation and Main Sequence★★★☆☆⏱ 8 min

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All stars form from the gravitational collapse of a cold molecular hydrogen nebula. As the cloud contracts, gravitational potential energy converts to thermal energy, raising core temperatures until hydrogen nuclei have sufficient kinetic energy to overcome electrostatic repulsion and initiate proton-proton fusion.

📘 Definition

Hydrostatic equilibrium

The stable balance between inward gravitational pressure and outward radiation pressure from core fusion that defines the main sequence phase.

Example:

Our Sun has remained in hydrostatic equilibrium for 4.6 billion years.

📐 Worked Example

Calculate the total power radiated by the Sun given its surface temperature of 5770 K and radius of $6.96 \times 1 0^{8}$ m.

1
Use the Stefan-Boltzmann law for luminosity:
2
$L = 4 \pi R^2 \sigma T_s^4$
3
Substitute known values, where $σ = 5.67 \times 1 0^{- 8} W m^{- 2} K^{- 4}$ :
4
$L = 4 \pi (6.96 \times 10^8)^2 \times 5.67 \times 10^{-8} \times (5770)^4$
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Simplify to get the standard solar luminosity value:
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$L = 3.85 \times 10^{26} W$

✓ Quick check

Confirm your understanding of main sequence properties:

What two forces balance to create hydrostatic equilibrium?
- Nuclear force and gravity
- Radiation pressure and gravity
- Electrostatic force and radiation pressure
- Nuclear force and electrostatic force
Reveal answer
Radiation pressure and gravity —
This balance prevents the star from collapsing or exploding during its stable lifetime.

Exam tip:

IB exam mark schemes almost always award 1 mark for explicitly stating hydrostatic equilibrium when describing main sequence stars.

2. Hertzsprung-Russell (HR) Diagram Classification★★★★☆⏱ 7 min

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The HR diagram plots stellar luminosity on the y-axis against surface temperature on the x-axis, with temperature increasing from right to left. 90% of all observable stars fall along the diagonal main sequence band, with distinct separate regions for red giants, supergiants, and white dwarfs.

HR Region	Temperature Range	Relative Luminosity	Stellar State
Main Sequence	2500 K - 40000 K	0.001 - 10000 $L_{⊙}$	Core hydrogen fusion
Red Giants	3000 K - 5000 K	10 - 1000 $L_{⊙}$	Shell hydrogen fusion
Supergiants	3000 K - 50000 K	10000 - 1000000 $L_{⊙}$	Multi-shell heavy element fusion
White Dwarfs	8000 K - 100000 K	< 0.01 $L_{⊙}$	No fusion, remnant core cooling

📐 Worked Example

Classify a star with surface temperature 4000 K and luminosity 100 times that of the Sun.

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Step 1: Locate 4000 K on the HR x-axis, which falls in the cool red star range.
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Step 2: Locate 100 $L_{⊙}$ on the y-axis, which is far above the top edge of the main sequence band.
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Step 3: Match the coordinates to the red giant region of the HR diagram.

3. Magnitude and Luminosity Calculations★★★★☆⏱ 6 min

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🔬 Derivation

Goal:

Derive the magnitude difference relation from apparent brightness

Starting from:

The magnitude scale is defined so that a 5 magnitude difference equals a 100x brightness ratio.

1
For a 5 magnitude difference: $\frac{b _{1}}{b _{2}} = 100$ when $m_{2} - m_{1} = - 5$
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Take base-10 logarithm of both sides: $lo g_{10} (b_{1} / b_{2}) = 2$
3
Rearrange to isolate magnitude difference: $m_{2} - m_{1} = - 2.5 lo g_{10} (b_{2} / b_{1})$

Result:

This logarithmic relation is the core formula for all IB magnitude calculation questions.

📐 Worked Example

Star A has apparent magnitude 1.2, Star B has apparent magnitude 4.7. Calculate the ratio of their apparent brightness.

1
Compute the magnitude difference: $m_{B} - m_{A} = 4.7 - 1.2 = 3.5$
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$3.5 = -2.5 \log_{10}(b_B / b_A)$
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Rearrange to isolate the brightness ratio:
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$\log_{10}(b_B / b_A) = -1.4$
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$b_B / b_A = 10^{-1.4} = 0.04$
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Final result: Star A is 25 times brighter than Star B.

4. End of Stellar Life: Low vs High Mass Stars★★★★★⏱ 9 min

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Methods compared

Stellar final remnant type depends entirely on the initial mass of the star:

Low mass star (< 8 $M_\odot$)

Fuses hydrogen then helium in core, forms planetary nebula, leaves behind a white dwarf remnant below the 1.4 $M_{⊙}$ Chandrasekhar limit supported by electron degeneracy pressure.

+ Pros: Stable remnant that cools slowly over trillions of years

− Cons: No supernova event occurs at the end of its life

High mass star (> 8 $M_\odot$)

Fuses heavy elements up to iron in core, undergoes core collapse supernova, leaves behind a neutron star below 2-3 $M_{⊙}$ Oppenheimer-Volkoff limit, or a black hole if the remnant exceeds this threshold.

+ Pros: Distributes heavy elements into interstellar space for new star formation

− Cons: Short total lifetime of only a few million years

📘 Definition

Schwarzschild radius

$R_{s} = \frac{2 GM}{c ^{2}}$

The radius of the event horizon of a non-rotating black hole.

Example:

The Schwarzschild radius for a 1 solar mass black hole is ~3 km.

📐 Worked Example

Classify the final remnant of a 20 solar mass star that leaves a 3.5 solar mass core after supernova.

1
Step 1: The initial 20 $M_{⊙}$ star is above the 8 $M_{⊙}$ threshold for high mass stars.
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Step 2: The 3.5 $M_{⊙}$ core exceeds the 2-3 $M_{⊙}$ Oppenheimer-Volkoff limit for stable neutron stars.
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Step 3: The remnant will collapse to form a stellar mass black hole.

5. Common Pitfalls

Wrong move:

Confusing apparent and absolute magnitude

Why:

Mixing up distance-dependent observed brightness and intrinsic stellar properties

Correct move:

Always note absolute magnitude is defined at a standard 10 parsec distance before solving any calculation

Wrong move:

Placing red giants on the main sequence

Why:

Forgetting main sequence stars fuse hydrogen in the core, while red giants fuse hydrogen in an outer shell

Correct move:

Label HR diagram axes explicitly before classifying any stellar population

Wrong move:

Using a linear scale for magnitude differences

Why:

Treating magnitude as a linear rather than logarithmic brightness scale

Correct move:

Use the 2.512^(Δm) brightness ratio rule for all magnitude difference calculations

Wrong move:

Stating low mass stars end as neutron stars

Why:

Ignoring the 1.4 solar mass Chandrasekhar limit threshold for white dwarf stability

Correct move:

Explicitly reference the mass limit when classifying stellar remnants to earn full exam marks

Wrong move:

Ignoring the inverse square law for apparent brightness

Why:

Forgetting observed brightness drops with the square of distance from the observer

Correct move:

Write the full $b = L / (4 π d^{2})$ formula before substituting any numerical values

6. Quick Reference Cheatsheet

Quantity	Formula	IB Required Constant
Stellar Luminosity	$L = 4 π R^{2} σ T_{s}^{4}$	$σ = 5.67 \times 1 0^{- 8} W m^{- 2} K^{- 4}$
Apparent Brightness	$b = \frac{L}{4 π d ^{2}}$	1 parsec = 3.26 light years
Magnitude Difference	$m_{2} - m_{1} = - 2.5 lo g_{10} (\frac{b _{2}}{b _{1}})$	Absolute M defined at d=10 pc
Chandrasekhar Limit	$1.4 M_{⊙}$	Maximum stable white dwarf mass
Schwarzschild Radius	$R_{s} = \frac{2 GM}{c ^{2}}$	Event horizon radius for black holes

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2024 · Paper 2
Stellar evolution 6-mark question
2023 · Paper 3
HR diagram classification task
2022 · Paper 2
Magnitude difference calculation

What's Next

Now that you have mastered core stellar properties and lifecycles, you are ready to apply these concepts to high-weight IB Physics HL Paper 2 and Paper 3 questions covering cosmic expansion and Hubble’s law. You will build on your understanding of magnitude scales to solve for interstellar distances using standard candles, a common 6+ mark extended response task. Stellar astrophysics is a foundational topic for all further cosmology content, so ensure you have practiced all magnitude and HR diagram classification questions before moving on. The linked modules below will deepen your mastery of the full Theme E astrophysics syllabus.