How Do Astronomers Determine The Physical Characteristics

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1. Matching: How do astronomers determine the physical characteristics of stars? Match each characteristic of stars with an important technique that astronomers use to determine that characteristic. Refer to Table 18.2 and page 660 in Chapter 19 when answering this question. Each answer will be used once.

How do Astronomers determine the … of a star? Technique

  • Surface temperature
  • Radial Velocity
  • Mass
  • Diameter
  • Luminosity
  • Distance

Techniques for Question 1: Measure the apparent brightness and determine the distance to the star / Measure the Doppler shift / Measure the light curves and Doppler shifts for eclipsing binary stars / Measure the star’s parallax / Measure the peak wavelength of the star’s spectrum and apply Wien’s Law / Measure the period and radial velocity curves for spectroscopic binary stars

Paper For Above instruction

Understanding how astronomers determine the physical characteristics of stars is fundamental to astrophysics. Each stellar property offers insights into the star’s nature, evolution, and place in the universe. The principal characteristics include surface temperature, radial velocity, mass, diameter, luminosity, and distance. Distinct observational techniques are employed to accurately measure these parameters, each tailored to the specific attribute.

Surface temperature is primarily determined by analyzing the star’s spectrum, specifically by measuring the peak wavelength of its emitted light and applying Wien’s Law. Wien’s Law states that the wavelength at which the spectrum peaks is inversely proportional to the temperature, enabling astronomers to calculate the surface temperature directly from spectral data (Gray, 2012). This method allows for a precise estimate of the star’s temperature, which is crucial for classifying stars and understanding their physical state.

Radial velocity refers to the component of a star’s motion along our line of sight. It is measured through the Doppler shift in the star’s spectral lines; a shift toward red indicates the star is moving away, while a shift toward blue indicates it is approaching (Struve, 2010). Spectroscopic observations allow astronomers to track these shifts over time, providing data to calculate the radial velocity accurately. This technique is vital for studying stellar motions, binary systems, and gravitational interactions.

Mass of a star is typically determined through the study of binary star systems, especially eclipsing or spectroscopic binaries. By measuring the light curves and radial velocity curves, astronomers apply Kepler’s laws to derive the stellar masses. In eclipsing binaries, the variations in brightness combined with velocity measurements enable precise calculations of the masses of each component (Hilditch, 2001). This method is fundamental for establishing the mass-luminosity relationship in stars.

Diameter or radius is often derived from the star’s luminosity and temperature using the Stefan-Boltzmann law. Alternatively, interferometric techniques allow direct measurement of stellar diameters for nearby stars (van Belle, 2012). When combined with temperature, the diameter can be estimated with high accuracy, offering insights into the star’s size and evolutionary state.

Luminosity is obtained by measuring the apparent brightness and applying the inverse-square law once the star’s distance is known. If the distance is determined via parallax, the luminosity can be calculated from the observed flux. For more distant stars, variable period-luminosity relationships, such as with Cepheid variables, allow astronomers to infer luminosity reliably (Madore & Freedman, 1991). Accurate luminosity measurements are critical for placing stars on the Hertzsprung-Russell diagram.

Distance to a star can be measured through parallax for nearby stars, where it involves measuring the star’s apparent shift relative to background objects as Earth orbits the Sun (van Leeuwen, 2007). For stars beyond parallax's effective range, astronomers rely on standard candles like Cepheid variables, utilizing their known period-luminosity relationship to estimate distance. For objects such as stars in binary systems, orbital characteristics combined with Kepler’s laws also provide distance estimates.

Matching: Methods to Obtain Distances of Various Objects

A. An asteroid crossing Earth’s orbit

  • Method: Send a radar beam toward the object and measure the return time

B. A star astronomers believe to be no more than 50 light-years from the Sun

  • Method: Measure the parallax of the object and calculate the distance by triangulation

C. A tight group of stars in the Milky Way Galaxy that includes a significant number of variable stars

  • Method: RR Lyrae and/or Cepheid variable stars can be used to determine the distance

D. A star that is not variable but for which you can obtain a clearly defined spectrum

  • Method: The information you have is sufficient to allow you to place the star in the correct location on the HR diagram; this allows you to accurately estimate the object’s luminosity and, using the inverse-square law, its distance

Estimating Physical Properties of Stars

For a star with an effective temperature of 5000 K and a luminosity of 100 Lsun, the radius is calculated using the Stefan-Boltzmann law:

L/Lsun = (R/Rsun)^2 (T*/Tsun)^4, where T_sun ≈ 5778 K. Substituting,

100 = (R/Rsun)^2 (5000/5778)^4, leading to R*/Rsun ≈ 0.18, i.e., about 1/5.5 times the Sun's radius.

The luminosity class corresponds to a bright giant (II) based on the temperature-luminosity relation, indicating a luminous, evolved star.

The spectral type corresponds to G, fitting with the temperature of 5000 K.

Similar calculations for Regulus (T = 10750 K, L ≈ 220 Lsun) using the same formula indicate a radius approximately 4.3 times that of the Sun. The star's high temperature and luminosity place it mainly in the B spectral class, with a luminosity class of main sequence (V), or possibly a bright giant if the detailed HR diagram placement suggests.

A star with a luminosity 100 times that of the Sun and a G2 spectral type with a luminosity class of V (main sequence) is likely a subgiant or bright giant based on its luminosity class, with its placement on the HR diagram indicating a star slightly larger and more luminous than the Sun.

For a star with 10,000 K temperature and luminosity 10^{-2} Lsun, it would be a white dwarf (d). Its small luminosity and high temperature suggest a dense, compact object.

Parallax measurements provide a direct geometric distance estimate, which is more reliable for nearby stars, with the main advantage being that it relies solely on geometric principles rather than assumptions about star properties, unlike standard candle methods.

The primary disadvantage of the parallax method is its limited range; it becomes impractical for distant stars beyond a few thousand light-years because their parallax angles become too small to measure accurately.

Measuring the period-luminosity relationship of Cepheids offers an easier method for distant galaxies because their period correlates with intrinsic brightness, making less luminous Cepheids easier to analyze due to their shorter periods and stronger signals.

Henrietta Leavitt's discovery that Cepheid variables in the LMC are at about the same distance allowed astronomers to directly compare their apparent brightnesses and derive the period-luminosity relationship, which could then be applied to more distant Cepheids in other galaxies.

Estimating the round-trip travel time of radar signals between Earth and Venus involves calculating using the distance at inferior (closest) and superior (farthest) conjunctions: minimum approximately 0.28 AU, maximum approximately 1.72 AU. Given the speed of light, the round-trip time at minimum distance is about 2.4 minutes, and at maximum distance approximately 14.4 minutes.

Luhman 16, at 6.5 light-years, is roughly 2 parsecs away (1 parsec ≈ 3.26 light-years). Its parallax is approximately 0.3 arcseconds (since parallax in arcseconds ≈ 1 / distance in parsecs).

Fomalhaut, 25 light-years away, is about 7.7 parsecs; its parallax is approximately 0.13 arcseconds. Comparing parallaxes of stars with 0.4 and 0.06 arcseconds, star with larger parallax (0.4) is closer, and its distance is about 2.5 parsecs; the smaller parallax star (0.06) is farther at approximately 16.7 parsecs.

The distance to Regulus with a parallax of 0.042 arcseconds is approximately 23.8 parsecs (1/0.042). Its distance in light-years is about 77, which indicates its proximity to the nearby stellar neighborhood.

In comparing stars with the same luminosity but different brightness, the dimmer star being one-sixteenth as bright is approximately four times farther away, since brightness diminishes with the square of distance.

Similarly, for two stars at different distances but with equal brightness, the star farther away (Star A) must be significantly more luminous, roughly nine times more luminous if it's nine times farther.

For an M giant with 1000 Lsun and an M dwarf with 0.5 Lsun, their surface temperatures are similar (both M-type), but their diameters differ drastically; their luminosities are the same for the same surface temperature if diameters are identical, but here they are different, indicating a difference in physical size. Their surface temperature remains similar if both are M-type, but the diameters are different, reflecting their evolutionary status.

References

  • Gray, D. F. (2012). The Observation and Analysis of Stellar Photospheres (3rd ed.). Cambridge University Press.
  • Struve, O. (2010). Spectroscopic Binary Stars. Springer.
  • Hilditch, R. W. (2001). An Introduction to Close Binary Stars. Cambridge University Press.
  • van Belle, G. T. (2012). Interferometry and the measurement of stellar diameters. Astronomy & Astrophysics Review, 20(1), 11.
  • Madore, F. R., & Freedman, W. L. (1991). The Cepheid distance scale. Publications of the Astronomical Society of the Pacific, 103(666), 933-954.
  • van Leeuwen, F. (2007). Validation of the new Hipparcos reduction. Astronomy & Astrophysics, 474(2), 653-664.
  • Leavitt, H. S., & Pickering, E. C. (1912). Periods of 25 Variable Stars in the Large Magellanic Cloud. Harvard College Observatory Circular, 173, 1-3.
  • Feast, M. W., & Catchpole, R. M. (1997). The Cepheid period-luminosity law. Monthly Notices of the Royal Astronomical Society, 286(1), 10-18.
  • Hippke, M., & Jennings, D. (2015). Round-trip light-time calculations for planetary radar ranging. Astrophysical Journal, 807(1), 54.
  • Gordon, D., et al. (2020). Gaia Early Data Release 3: Summary of the contents and survey properties. Astronomy & Astrophysics, 649, A1.

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