1. Explain the difference between apparent brightness and intrinsic brightness, and know what the terms apparent magnitude, absolute magnitude, and luminosity mean.

2. Describe how the luminosity of a star of a specific surface temperature depends on its diameter.

3. Describe how trigonometric parallax is used to determine a star’s distance.

4. Describe how the inverse square law of light can be used to determine the intrinsic brightness (or absolute magnitude or luminosity) of a star.

5. Discuss how an HR diagram is constructed.

6. Describe the method of spectroscopic parallax for determining the distance to a star.

7. Discuss stellar motions and how they are measured.

¶ intrinsic brightness

¶ luminosity

¶ stellar diameter

¶ perceived or apparent brightness

¶ absolute magnitude

¶ apparent magnitude

¶ stellar distance

¶ trigonometric parallax

¶ arc sec

¶ inverse square law

¶ spectral type

¶ Hertzsprung-Russell (HR) Diagram

¶ main sequence

¶ proto-stars

¶ giant and supergiant stars

¶ white dwarfs and neutron stars

¶ spectroscopic parallax

¶ Doppler effect

¶ stellar motion

¶ radial velocity

¶ transverse velocity

¶ proper motion

¶ stars’ orbits about the Galaxy

¶ The total amount of energy emitted by a star per second is a measure of the star’s actual or intrinsic brightness. This is often referred to as the star’s luminosity (L).

¶ The luminosity (L) of a star depends on its surface temperature (T) and its diameter (D). The Stefan-Boltzmann Law says that a lot more energy per second per m² (E) will come out of an area on the surface of a star if you increase its temperature a little bit (E = sT⁴ where s stands for the constant, "sigma."). If the diameter of a star is increased, a lot more total energy per second (luminosity) will come out, because you are increasing the surface area of the star--there's more area available to emit light, now. Multiplying the Stefan-Boltzmann formula for energy per second per surface area by the formula for surface area in terms of diameter, (pi D²) we find the total luminosity to be (L = pi D² sT⁴). The important part of this is to remember that Luminosity is proportional to D² and proportional to T⁴.

¶ The apparent or perceived brightness of a star is different than its actual or intrinsic brightness. This is because a star of fixed luminosity would become fainter if we could move it further away.

¶ Thus, the apparent or perceived brightness of a star depends on the star’s actual or intrinsic brightness (luminosity) and the star’s distance.

¶ The relation between apparent brightness and distance is governed by the inverse square law: the apparent brightness is inversely proportional to the distance to the object squared. [For example, if we were twice as far from the Sun, the Sun would appear 2² = 4 times fainter; if we were 10 times as far from the Sun, the Sun would appear 10² = 100 times fainter.]

¶ Absolute magnitude is a unit of measure that astronomers use to specify the actual or intrinsic brightness (luminosity) of a star. Note that the actual or intrinsic brightness (luminosity) of a light bulb is given by its wattage.

¶ Apparent magnitude is a unit of measure that astronomers use to specify the apparent or perceived brightness of a star. The apparent magnitude of a star depends both on the star’s absolute magnitude and the star’s distance.

¶ The magnitude scale was invented by ancient astronomers and, unfortunately, is still in use today. If the magnitude of a star decreases by 1 unit, that corresponds to the star being about 2.5 times brighter. If the magnitude of a star decreases by 5 units, that corresponds to the star being 100 times brighter.

[For example, the Sun has an apparent magnitude of -26.8; the Moon has an apparent magnitude of -12.6; Venus at its maximum brightness has an apparent magnitude of -4.4; Sirus, the brightest star, has an apparent magnitude of -1.4; the faintest stars we can see with the unaided eye have apparent magnitudes of 6.]

¶ Photometry is the technique astronomers use to measure the apparent brightness (or apparent magnitude) of an object. Usually the measurement is made with a filter that only allows light corresponding to a very specific set of wavelengths to be measured. By making several measurements in different filters (e.g., a blue filter and a red filter), a star’s color can be measured.

¶ The most fundamental method astronomers use to measure the distances of nearby stars is trigonometric parallax.

¶ Trigonometric parallax refers to half the angle through which a star appears to be displaced (in relation to extremely distant stars) as the Earth moves from one side of the Sun to the other.

¶ The distance (d) to a star in parsecs is: d = 1/p where the parallax (p) is in seconds of arc. One parsec is about 2 x 10⁵ astronomical units (an astronomical unit is the distance between the Earth and Sun) or 9.5 x 10¹² km or 3.26 light years. Note that a circle has 360 degrees; 1 degree has 60 minutes of arc; and 1 minute of arc has 60 seconds of arc.

¶ Once the distance to a star is known, it can be used along with the apparent or perceived brightness of the star to determine the star’s actual or intrinsic brightness (luminosity). The inverse square law is used to do this calculation.

¶ If we wanted to say this using astronomical ‘jargon’ we would say that the absolute magnitude of a star can be determined from its apparent magnitude and parallax.

¶ So far we have discussed three fundamental stellar properties which can be determined from observations:

¶ The Hertzsprung-Russell Diagram (sometimes called the HR diagram) is formed by making a plot or graph of:

¶ Making HR diagrams is the most fundamental tool that astronomers have for studying the properties and evolution of stars.

¶ In very general terms, stars fall in three areas on the HR diagram:

¶ At some point, a star becomes so distant that trigonometric parallax cannot be used to accurately measure the distance between the Earth-Sun system and the star.

¶ When this happens, spectroscopic parallax can be used to determine a star’s distance.

¶ The method of spectroscopic parallax relies on using a star’s spectrum (color and spectral lines) to locate the position of the star on the HR diagram. Once this is done, the star’s luminosity (or absolute magnitude is also known. The star’s apparent brightness (or apparent magnitude) is observed using photometry. The inverse square law can then be used to calculate the star’s distance.

¶ The Doppler Effect is a change in wavelength that results when a source of waves and the observer are moving relative to each other. The Doppler Effect is one of the most important tools that astronomers can use to understand the Universe, because it can be used to measure the speed at which an object is moving toward or away from us.

¶ When a wave is emitted by an object moving toward you, the wavelength is seen to decrease. Similarly, when a wave is emitted by an object moving away from you, the wavelength is seen to increase. This is the Doppler Effect. The amount of increase or decrease in wavelength allows one to calculate the velocity difference between you and the object.

¶ With electromagnetic radiation, the amount of the shift in wavelength depends on how fast an object is moving compared to the speed of light. Something moving 3,000 km/s will produce a 1% shift in wavelength since the speed of light is 100 times greater.

¶ In our everyday experience we often hear the Doppler Effect involving sound waves. For example, the change in pitch of a passing train. Note that sound waves are something very different from electromagnetic radiation. We hear as a result of sound waves (rapid changes in atmospheric pressure), but we can’t hear a radio wave. Radio waves are detected using antennas; they are interpreted and converted to sound waves using a receiver, amplifier, and speaker.

¶ Just as the planets are in motion around the Sun, there are other motions throughout the Galaxy and the Universe.

¶ For example, binary stars are in orbit around one another. Furthermore, the 100 billion stars which comprise the Milky Way Galaxy are in orbit about the Galactic center.

¶ Stellar motions can be observed in two ways:

¶ The radial velocity and transverse velocity of a star can be used to determine a star’s space velocity or orbit in the Galaxy.

¶ By analyzing the properties of stars’ orbits in the Galaxy, we learn about the mass of the Galaxy because mass gives rise to the gravity that determines stars’ orbits. We learn about both the visually luminous and visually non-luminous mass by studying these motions, because both kinds of mass have gravity which affects the motions of objects.