Properties of Stars Study Notes

ASTR 1P02 - Lecture 13: Properties of Stars

Overview of Topics Covered

• Brown dwarfs
• Stellar masses and their relation to luminosity
• The Hertzsprung–Russell (H-R) diagram
• Luminosity classes
• Various types of stars
• Measuring distances to the stars
• Additional information
  Image: A star-forming region in the Large Magellanic Cloud. Credits: ESA/Hubble

Stellar Masses

• Stars exhibit a very wide range of masses.
• Stellar masses are typically measured in units of solar mass, denoted as 𝑀⊙.
  • 𝑀⊙ ≈ 2 × 10³⁰ kg.
• The smallest stars have a mass of approximately 𝑀⊙/12. Below this threshold, nuclear fusion of protons into helium to produce light is not possible.

Brown Dwarfs

• Objects with masses from approximately 𝑀⊙/80 to 𝑀⊙/12 cannot achieve proton fusion like true stars but can fuse deuterium, an isotope of hydrogen.
• Deuterium Fusion: This is the second stage of the proton-proton chain.
• These objects are referred to as brown dwarfs, also termed as “failed stars” due to their inability to ignite proton fusion.
• Brown dwarfs are not necessarily brown in color; their appearance can range from orange to red when warm, and can appear magenta or black when cold.
• The mass range of brown dwarfs is between approximately 13 to 80 𝑀𝐽, where:
  • 𝑀𝐽 ≈ 2 × 10²⁷ kg (mass of Jupiter).
  • Brown dwarfs are therefore less massive than stars but more massive than planets because they generate a small amount of light through nuclear reactions.

Comparison of Masses

• Brown dwarfs are slightly larger than Jupiter and can be up to 80 times more massive, resulting in higher density compared to Jupiter.
  • Image Credit: Planetkid32 (Wikipedia)

Spectral Classes of Brown Dwarfs

• Stellar classification includes spectral classes that rank from hottest to coldest: O, B, A, F, G, K, and M.
• The hottest brown dwarfs belong to class M while the colder variants fall under additional spectral classes: L, T, and Y.
  • Brown dwarfs emit most of their light in the infrared spectrum due to low temperatures.
  • Infrared light has lower frequency compared to red light, indicating lower energy and colder temperatures.

Properties of Various Stars

• The relationship between mass and luminosity shows that more massive stars are also more luminous.
  • The equation that describes this relationship is: 𝐿 ∝ 𝑀⁴
  • Meaning if the star's mass increases by a factor of 2, its luminosity increases by a factor of 2⁴ = 16.
  • This exponent of 4 holds for stars with masses between 0.4 and 2 𝑀⊙, though the exponent may vary outside this mass range.

Hertzsprung–Russell (H-R) Diagram

• The H-R diagram plots stars based on two defining properties:

  1. Horizontal Axis: Temperature (or spectral class); hotter stars are portrayed to the left.
  2. Vertical Axis: Luminosity (or absolute magnitude); brighter stars are shown towards the top.

• When stars are plotted on the H-R diagram, notable clustering into regions is observed.

  • Most stars reside along the main sequence, which runs diagonally from the upper left (bright and hot) to the lower right (dim and cold).
  • Hotter temperatures correlate with greater luminosity for main sequence stars.

Star Types in the H-R Diagram

• Giants: Found on the upper right, these possess low temperatures but high luminosities due to their vast surface areas.
• White Dwarfs: Located on the bottom left, they exhibit high temperatures yet low luminosities, indicating their small size compared to giants.

Luminosity Classes on the H-R Diagram

• Stars are classified into luminosity classes on the H-R diagram:
  • Ia+: Hypergiants (extremely luminous supergiants)
  • Ia: More luminous supergiants
  • Ib: Less luminous supergiants
  • II: Bright giants
  • III: Normal giants
  • IV: Subgiants
  • V: Main sequence stars; sometimes referred to as “dwarfs”
  • VI: Subdwarfs
  • VII: White dwarfs
• The notation for spectral and luminosity classes is combined (e.g., B2V indicates a main-sequence B2 star).

Characteristics of Dwarfs and Stars

• Terminology surrounding “dwarfs” can be confusing as not all main sequence stars are smaller than giants, while brown or white dwarfs are not classified within the main sequence.
• Dwarf Classifications by Color:
  • Red Dwarfs (Class MV): Smallest and dimmest main-sequence stars.
  • Typical temperature: ~2,000-3,900 K, mass: ~0.07-0.6 𝑀⊙.
  • Example: Proxima Centauri (M5.5V).
  • Orange Dwarfs (Class KV): Temperature: ~3,900-5,300 K, mass: ~0.6-0.9 𝑀⊙; Example: Alpha Centauri B (K1V).
  • Yellow Dwarfs (Class GV): Temperature: ~5,300-6,000 K, mass: ~0.9-1.1 𝑀⊙; not actually yellow but white in appearance; Example: The Sun (G2V).
  • Yellow-White Dwarfs (Class FV): Temperature: ~6,000-7,600 K; mass: ~1.1-1.4 𝑀⊙.
  • A-type Dwarfs (Class AV): Temperature: ~7,600-10,000 K, mass: ~1.4-2.1 𝑀⊙; Example: Vega (A0V).
  • B-type Main-Sequence Stars (Class BV): Temperature: ~10,000-30,000 K, mass: ~2.1-16 𝑀⊙; Example: Algol A (B8V).
  • O-type Main-Sequence Stars (Class OV): Temperature: ~30,000-50,000 K, mass: ~16-90 𝑀⊙; Example: 10 Lacertae (O9V).

Giants and Their Classification

• Giants can be further classified into:
  • Red Giants: Coldest temperatures (Spectral Classes K and M); Example: Arcturus (K1.5III).
  • Yellow Giants: Intermediate temperatures (G, F, A); Example: Sigma Octantis (F0IV).
  • Blue Giants: Hottest temperatures (O, B); Example: Alcyone (B5III).

Statistical Breakdown of Stars

• Approximately 90% of stars fall within the main sequence category.
• Roughly 10% are classified as white dwarfs.
• Less than 1% are categorized as giants.
• The distribution of stars correlates with their lifecycle stages. Stars form from dust clouds collapsing under their own gravity and spend about 90% of their lifetimes in the main sequence stage, transitioning to giants and white dwarfs in later stages.

Celestial Distances

• Distances to nearby objects can be measured directly (e.g., distance to the Moon ~384,000 km using laser reflection).
  • The speed of light (~300,000 km/s) allows calculation of distance based on the return time of the laser beam.
• Lunar Ranging Retro Reflector (LRRR) left by Apollo 15: Enables accurate distance measurements up to 1 mm.
• Stars, being immensely farther away, require indirect methods for distance measurement.

Indirect Measurement Methods

Stellar Parallax

• Stellar parallax, covered in previous lectures, allows spatial comparison by observing a star's position shifts compared to more distant stars against the backdrop.
  • The parallax angle 𝑃 reflects the angle formed using 1 Astronomical Unit (AU) as the baseline, which corresponds to half the Earth's orbital diameter.
• Parallax Measurement:
  • 1 degree = 3,600 arcseconds.
  • A star with a parallax of 1 arcsecond is approximately 3.26 light-years away, known as a parsec (pc).
  • Basic Formula: Distance in parsecs = 1 / Parallax in arcseconds.

Example Problem

• Given a star with a parallax of 0.5 arcseconds, the distance to the star is:
  • A: 5 parsecs
  • B: 0.5 parsecs
  • C: 2 parsecs
  • Correct Answer: C: 2 parsecs

Summary of Distance Units in Astronomy

• Meter: Based on the distance light travels in a fraction of a second.
• Kilometer (km): 1 km = 1,000 m.
• Light-year (ly): Distance light travels in one year.
• Parsec (pc): Distance to a star with a parallax of 1 arcsecond.
  • 1 light-year ≈ 0.31 pc ≈ 9.46 trillion km.
  • 1 parsec ≈ 3.26 light-years ≈ 30.9 trillion km.

Further Simulation

• A simulation that illustrates the effects of stellar parallax at varying distances can be found at the provided URL.

Limitations of Stellar Parallax

• Stellar parallax is limited to relatively nearby stars due to the maximum observable distance of ~30,000 light-years, where Gaia space observatory is currently measuring.
• The Milky Way's radius is approximately 87,000 light-years, positioning many stars beyond the range where parallax is measureable.

Inverse-Square Law of Light

• The brightness of stars diminishes by the square of their distance from an observer.
• Inverse-Square Law Equation: If stars had the same luminosity, apparent brightness differences could be used to determine distances.
• However, different stars possess varied luminosities as seen from the H-R diagram.

Standard Candles

• Standard Candles: Types of stars with definable and consistent luminosity, allowing distance calculations by comparing brightness.
• Variable Stars: A notable example includes Cepheid variables, whose periodic brightness changes can be tracked and measured.

Cepheid Variables

• Definition: Yellow bright giants and supergiants of spectral class F6-K2 with masses of ~4-20 𝑀⊙ and luminosities of ~100-1,000 𝐿⊙, varying every ~3-50 days.
• Their light curves illustrate fluctuations over specific time frames.
• Discovered in nearby galaxies (Large and Small Magellanic Clouds) where all stars can be assumed to be at similar distances, allowing luminosity determination through brightness comparisons.
• Henrietta Leavitt established the period-luminosity relationship, establishing that longer periods correspond to greater luminosity.

Using Cepheids as Standard Candles

• Once the luminosity is known from the period, it can be correlated with apparent brightness to estimate distance using the inverse-square law, effectively utilizing Cepheids as standard candles.

RR Lyrae Variables

• RR Lyrae Variability: More common than Cepheid variables with shorter pulsation periods (~7-24 hours).
• These stars exhibit consistent brightness within star clusters, making them useful standard candles as they possess similar apparent brightness and luminosity (~40-50 𝐿⊙).

Conclusion of Distance Measuring Techniques

• By understanding the characteristics of stars and their classifications, astronomers can accurately measure distances using various established methodologies. The H-R diagram, standard candles, and distances discussed serve as foundational knowledge in stellar astronomy.

Cosmic Distance Ladder

• A summary of distance measuring techniques, including their effective ranges:
  • Stellar Parallax: Up to ~30,000 light-years.
  • RR Lyrae Variables: Up to ~300,000 light-years.
  • H-R Diagram: Up to ~1,200,000 light-years.
  • Cepheid Variables: Up to ~60,000,000 light-years.

Historical Context

• Prior to the 1920s, the Milky Way galaxy was considered the entirety of the universe. Edwin Hubble’s work with Cepheid variables revealed Andromeda's true distance (~2,500,000 light-years), establishing it as a separate galaxy.

Closing Remarks

• This lecture encompassed various star types, their classifications, and techniques for measuring astronomical distances.
• Further reading is recommended in OpenStax Astronomy, chapters 18-19, while practice exercises are available in the course materials.