Lecture 13: Stellar Properties and How We Measure Them

We are now moving to a domain where all stars are unresolved points of light, and we must use a variety of strategies to learn about them.

What Can Be Measured?

--

position, distance can be difficult, accurate determination of RA and DEC is called astrometry--

velocityvia Doppler shift or by looking for the slow change in RA and DEC with time called proper motion--

temperature, crudely by measuring brightnesses at enough wavelengths to determine , more precisely using spectroscopic techniques--

composition, using spectroscopic techniques--

massif the star is in a binary pair--

magnetic field, using spectroscopic techniques--

polarizationof starlight, may be indicative of interstellar dustDetermining distance is crucial because otherwise we cannot know what a star's intrinsic properties are. Once we have measured the distances to enough stars, we can use radiation laws to determine properties of other, more distant stars.

Direct Measurement of Distances

- relies on trigonometric parallaxes, use of space satellites has improved accuracy so that distances of up to 200 pc can be measured [but this is a small fraction of the size of the Milky Way]
- parsec unit of distance (=3.086x10
^{13}km) is defined by the parallaxes from Earth:

"=arc sec

- parallax is measured by observing a star when the Earth is on one side of the Sun and then 6 months later.

Measurement of Velocities

- Three velocity components need to be measured to specify a star's motion completely; radial velocity which gives the motion along the line-of-sight is the easiest.
- The transverse velocities in RA and DEC are more difficult to measure except for the closest stars. However, if data can be collected over a long time, more distant or smaller velocities determined. These proper motions can only be converted to physical velocities (rather than angular) if the distance to the star in known.

- Combining the transverse and radial velocities yields the star's true three-dimensional motion through space.
Photometry

recall that the flux that we measure at Earth from an object depends on its distance from us:

d= distance from Earth

MagnitudesFor expressing fluxes, astronomers use a system of magnitudes based on how the human eye responds to light (i.e., this is a historical system rather than a strictly logical system!). The mathematical definition of this system is

The logarithmic character of this relation and the factor of 2.5 relate to how the eye senses light and the interpretation in our brains.

For example, a flux ratio of 100 corresponds to a magnitude difference of 5:

Because a magnitude difference is the log of a ratio of fluxes, we must define a reference flux to convert a magnitude to a flux. This is usually done by declaring that m

_{1}is 0 and that flux_{1 }is the0-magnitude flux.The magnitude system is often used with filters to define a range of wavelengths (or color). Such systems usually include in their definition an average or effective wavelength to describe the filter and the convention that m

_{1}in the magnitude definition = 0 and the flux_{1}is specified and called the zero magnitude flux. One such system is the UBVRI (or Johnson system) where the flxues from the star Vega are used as the zero magnitude fluxes:

Filter Name Effective Wavelength (nm) 0-magnitude flux (Jy)

U 360 1880 B 440 4400 V 550 3880 R 700 3010 I 880 2430 This figure is similar to Figure 6.6 in the text.

The

effective wavelengthis the wavelength to which the filter measurement refers. For example, if you are told that the flux through the R filter is .058 Jy, then the star's flux at 700nm is .058 Jy.Example:

You observe a star whose V magnitude (denoted just as V) is +4.3. What is its flux in Jy?

Notice that magnitudes cannot be added to get the total flux from two or more objects. You must convert to flux, add the fluxes, and then convert back to magnitudes if the magnitude for the light from several objects is required.

Two types of magnitudes can be defined:

m = apparent magnitude = what you observe

M = absolute magnitude = what you would observe if the star were 10 pc distant (a measure of the star's luminosity)Note that M is frequently given with a subscript indicating which filter: M

_{V}means the absolute magnitude in the V filter passband. In this scheme, m is usually represented by the uppercase letter for the filter, e.g. V and m_{V }mean the same thing.These two magnitudes can be combined with distance:

m - M is called the

distance modulus.For example, Sirius has an absolute visual magnitude M

_{V }= 1.4 and a distance of 2.65 pc. What will its apparent m_{V}(usually denoted as V) be?V - M

_{V }= 5 log d -5V = 5 log (2.65) - 5 + 1.4 = -1.48

Another example:

A star has m=5.2 and M=4.3. How far away is it?

ColorsA color index is defined as the difference in magnitudes between two difference filters:

B - V = 2.5 log (f

_{v}/f_{B}) + constantThe constant can be computed from the 0 magnitude fluxes quoted for the photometric system.

B=-2.5log(f

_{B }/ 4400Jy)=-2.5log(f_{B }) + 2.5 log(4400Jy)V=-2.5log(f

_{V }/ 3880Jy)=-2.5log(f_{V }) + 2.5 log(3880Jy)so

B - V = -2.5log(f

_{B }) - -2.5log(f_{V })+ 2.5 log(4400Jy)-2.5 log(3880Jy)= 2.5log(f

_{v}/f_{B}) + 2.5log(4400Jy/3880Jy)= 2.5log(f

_{v}/f_{B}) + .137This color index is very handy for most stars as it is a convenient measure of the star's temperature:

B-V Temperature(° K) -0.32 47,000 0.00 9,410 0.31 7,160 0.59 6,010 0.66 5,780 1.15 4,270 1.61 3,260 Such a table must be calibrated observationally as stars are not perfect blackbodies:

(similar to Figure 6.7 in text)

Other colors can also be defined such as U-B.

Magnitudes and colors for the Sun:

Apparent Magnitude Absolute Magnitude ColorsU _{Sun = -25.85}M _{Usun}= +5.72B _{Sun = -25.85}M _{Bsun}= +5.54(U-B) _{Sun}= 0.18V _{Sun}= -26.70M _{Vsun}= +4.87(B-V) _{Sun}= 0.67Bolometric Magnitudes and Luminosities

Recall that

A magnitude representing luminosity can be defined and is called the bolometric magnitude. This magnitude is defined relative to the Sun:

where 4.74 = bolometric magnitude for the Sun.

The difference between M

_{bol}and M_{V}= B.C. , bolometric correction.