Julian Date: 2458995.16
2019-2020: CLV
THE DAILY ASTRONOMER
Wednesday, May 27, 2020
Remote Planetarium 42: Stellar Properties
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A profuse apology to the math phobes.
While we are designing these classes for the layperson, at times we truly need to delve into mathematics more than some people would like. Today is one of those times. While we promise we will be as conceptual as possible, we can't completely avoid the math. We will offer a conceptual summar at the article's conclusion.
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Today we begin to learn how we can determine stellar properties with the use of only a few pieces of information. First, we offer a quick review of terms
- Apparent magnitude (m) a measure of a celestial body's apparent brightness. In this system, the brighter the star, the lower the magnitude value. Invented by Hipparchus of Nicea and then quantified by N.R. Pogson, the magnitude system's original range extended from 1 - 6, with the brightest stars classified as magnitude 1 and the dimmest stars visible to the unaided eye classified as magnitude 6. The system has expanded considerably since its inception. Now, the range extends from -26.7 (the Sun's apparent magnitude) to 27, the faintest stars that are telescopically detectable. The magnitude scale is logarithmic with a factor of 2.5. So, a magnitude 1 star is 2.5 times brighter than a magnitude 2 star, which is 2.5 times brighter than a magnitude 3 star, et cetera. The actual factor is approximately 2.512, which equals the fifth root of 100. So, a magnitude 1 star is precisely 100 times brighter than a magnitude 6 star.
- Absolute magnitude (M) Absolute magnitude measures a celestial body's intrinsic brightness. The absolute magnitude value is equal to what the celestial body's magnitude would be were it 10 parsecs away.
- Light year: the distance that light travels in one Earth year when moving through a vacuum. As light speed in a vacuum equals 186,290 miles per second, a light year is equal to approximately 5.8 trillion miles.
- Luminosity: a radiant body's energy output per second. A star's luminosity is proportional to the square of its radius and the fourth power of its temperature.
- Parallax angle: the angle by which a foreground object's position shifts relative to more distant objects. Astronomers employ the parallax method to ascertain the distances of stars. The parallax method is valid out to a distance of 500 parsecs.
- Parsec: a unit of distance used in astronomy. A parsec equals approximately 3.26 light years.
The first task at hand is to determine a star's absolute magnitude, or its intrinsic brightness. Fortunately, determining this value requires only two pieces of information: the star's apparent magnitude and its distance. As we are still loitering around the general galactic vicinity, we shall restrict ourselves to the parallax determination method. Recall that to calculate a star's distance in parsecs, one need only divide 1 by the parallax angle expressed in arc-seconds:
d = 1/p"
Once we determine this value, we turn next to the
DISTANCE MODULUS:
m - M = 5log(d) - 5
where m = apparent magnitude; M = absolute magnitude; d = distance in parsecs. The log(d) is the logarithm of d to the base 10. (Refer to the rapid logarithm review posted at the end of this article.) We can rearrange this equation to solve for M:
M = m - 5log(d) + 5
m - M is known as the "distance modulus" To help with understanding, we will work through an example:
Ross 248:
Parallax angle: 0.316"
The distance equals 1/0.316" = 3.16 parsecs (approximately 10.3 light years)
The apparent magnitude is 12.3, a very faint star. It is more than one hundred times fainter than the dimmest stars visible to the unaided eye!
Now, we have the distance (3.16 parsecs) and the apparent magnitude (12.3).
We turn next to the distance modulus equation:
M = m - 5log(d) + 5
M = 12.3 -5log(3.16) + 5
log (3.16) = 0.499 (Or, 10 raised to the power of 0.499 equals 3.16. See the rapid logarithm review below.)
M = 12.3 -(5)(0.499) + 5
M = 12.3 -2.495 + 5
M = 14.8
If Ross 248 were at a distance of 10 parsecs, its magnitude would be 14.8. Quite faint, indeed. For comparison, if the Sun were at a distance of 10 parsecs, its magnitude would be 4.83
Now that we know Ross 248's absolute magnitude, we can now determine its luminosity relative to the Sun.
This time, we will avoid an equation and use a handy chart as seen below. Ross 248's absolute magnitude is 14.8. Its luminosity is 0.0018 that of the Sun. Here we need to point out that the luminosity is related to the star's bolometric magnitude, the magnitude averaged over the entire EM spectrum. We will discuss bolometric magnitudes in greater detail later in the course. Our aim today is to explain the procedure by which we can ascertain a star's luminosity
We can then calculate Ross 248's luminosity: about 0.18% that of the Sun's. A low luminosity star to be sure. Once we know the star's luminosity we can proceed to the next stage: its size and temperature. We recall the luminosity equation:
Since we know the star's luminosity relative to the Sun, we can set a ratio of the star's luminosity to that of the Sun. Setting this ratio eliminates the 4, pi and the Stefan-Boltzmann constant, leaving only the radius squared and the temperature raised to the fourth power. If we can determine the value of either the size or the temperature we can readily calculate the value of the other. Remotely, we cannot ascertain a star's size. We can determine a star's temperature. At some point soon we'll discuss temperature determination procedures in greater detail. Today's focus will be on the star's spectral type. By observing the star's spectrum, astronomers can classify it by spectral type. The main spectral classifications are as follows
The arrangement O B A F G K M is also a temperature sequence, with O stars being the hottest and M the coolest. Remember the mnemonic "Oh, be a fine girl kiss me!" Or, if you're worried about offending someone, you can replace "girl" with "gorilla," or "goldfish" or "gerontologist," if you happen to be fond of scientists who study aging.)
The associated temperatures are as follows:
- O 28,000 - 50,000 K*
- B 9,900 - 28,000 K
- A 7400 - 9900 K
- F 6000 - 7400 K
- G 4900 - 6000 K
- K 3500 - 4900 K
The different spectral classes are further divided into numerical gradations specific to more narrow temperature ranges.
Ross 248 is an M6 V star. We conclude that it's effective temperature is 2,799 K. (V indicates a "dwarf" or "main sequence" star, terms we'll introduce tomorrow.) We can then calculate the size once we know the luminosity and temperature. Ross 248's radius is 0.16 that of the Sun. Not only is Ross 248 a low luminosity star, it is also cool and small, at least by stellar standards. Were we to approach this star, ourselves, it would seem gargantuan!
CONCEPTUAL SYNOPSIS
- STEP 1: We can use trigonometric parallax to determine the distances to the closest stars. A nearby star's distance (in parsecs) equals 1 divided by the parallax expressed in arc-seconds.
- STEP 2: We use the distance modulus to determine the star's absolute magnitude. This modulus equation relates a star's absolute magnitude, apparent magnitude and distance. We can directly observe a star's apparent brightness and so will know its apparent magnitude. By knowing the star's apparent magnitude and distance, we can calculate the star's absolute magnitude or intrinsic brightness
- STEP 3: A star's absolute magnitude is directly related to its luminosity, or energy output. We can use a chart (as we did today) or a formula to calculate the star's luminosity relative to that of the Sun.
- STEP 4: We know that a star's luminosity is related to its size (radius) and temperature. By knowing a star's luminosity value relative to the Sun we can set up a ratio relating the star's size and temperature to that of the Sun, as well.
- STEP 5: By observing the star's spectral type, discernible through analysis of its spectrum, or light emissions, we can determine its temperature. The main spectral classes are O, B, A, F, G, K and M. This sequence is in order of decreasing temperature, with O being the hottest and M the coolest.
- STEP 6: Now that we know the luminosity and temperature, the star's size (radius) can be calculated.
Through these steps astronomers can measure the luminosities, sizes and temperatures of the close stars. As we will learn, these properties will lead us to know the stars' masses and even life cycles. Tomorrow we begin our discussion of the H-R Diagram: the main tool stellar astronomers employ to measure stars.
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Rapid logarithm review:
Before the advent of super scientific calculators that now enable people to perform immensely complex calculations with just a few button presses, mathematical people needed tools to assist them in their complicated computations. One of the most useful tools was the logarithm, invented in the early 17th century by the British astronomer/mathematician John Napier (1550-1617). Logarithms transformed intricate multiplication problems into more manageable addition problems.
Now, to understand them, let's examine a few examples:
Log 100 = ?
Most logarithms include a base number written in subscript between the word "log"and the number. If no number appears, base 10 is assumed. In the example above, Log 100 = is asking us, "10 raised to what number equals 100?" The answer is 2 because 10^2 (or 10 x 10) equals 100.
Log 100 = 2
Log 1000 = ?
"10 raised to the power of which number equals 1000?" We know that 10 raised to the 3rd power (or 10 x 10 x 10) = 1000.
Log 1000 = 3
One can insert any number next to the logarithm to yield a value, although not necessarily a neat whole number.
Log 23 = 1.36172. If you raise 10 to the power of 1.36172, the result will be 23.
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*K = Kelvin. An absolute temperature scale. 0 K is absolute zero, equal to -273.15 degrees C (-460 degrees F).
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