Basic Properties of Stars - 3 lLuminosities lFluxes lMagnitudes lAbsolute magnitudes.

Basic Properties of Stars - 3Basic Properties of Stars - 3Basic Properties of Stars - 3Basic Properties of Stars - 3

LuminositiesLuminosities

FluxesFluxes

MagnitudesMagnitudes

Absolute magnitudesAbsolute magnitudes

Solid AngleSolid AngleSolid AngleSolid Angle The The solid anglesolid angle, , , that an object subtends at a , that an object subtends at a point is a measure of how big that object appears point is a measure of how big that object appears to an observer at that point. For instance, a to an observer at that point. For instance, a small object nearby could subtend the same solid small object nearby could subtend the same solid angle as a large object far away. The solid angle angle as a large object far away. The solid angle is proportional to the is proportional to the surface areasurface area, , SS, of a , of a projection of that object onto a projection of that object onto a spheresphere centered centered at that point, divided by the square of the at that point, divided by the square of the sphere's radius, sphere's radius, RR. (Symbolically, . (Symbolically, = k S/R²= k S/R², where , where kk is the proportionality constant.) A solid angle is the proportionality constant.) A solid angle is related to the surface area of a sphere in the is related to the surface area of a sphere in the same way an ordinary same way an ordinary angleangle is related to the is related to the circumferencecircumference of a of a circlecircle.If the proportionality .If the proportionality constant is chosen to be 1, the units of solid constant is chosen to be 1, the units of solid angle will be the angle will be the SISI steradiansteradian (abbreviated (abbreviated srsr). ). Thus the solid angle of a sphere measured at its Thus the solid angle of a sphere measured at its center is 4center is 4 sr, sr,

For FunFor FunFor FunFor Fun

1) What is the angular size of the Sun 1) What is the angular size of the Sun as seen from Earth?as seen from Earth?

Radius Sun = 7.0 x 10Radius Sun = 7.0 x 1055 km km Distance to Sun = 1.5 x 10Distance to Sun = 1.5 x 1088 km km

2) What is the solid angle of the Sun 2) What is the solid angle of the Sun as seen from Earth?as seen from Earth?

3) What fraction of the sky does the 3) What fraction of the sky does the disk of the Sun then cover?disk of the Sun then cover?

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Luminosities and magnitudes of starsLuminosities and magnitudes of starsLuminosities and magnitudes of starsLuminosities and magnitudes of stars

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Luminosities and magnitudes of starsLuminosities and magnitudes of stars Luminosities and magnitudes of starsLuminosities and magnitudes of stars

Consider some source of radiationConsider some source of radiation

Intensity IIntensity I = energy emitted at some = energy emitted at some

frequency frequency , per unit time dt, per , per unit time dt, per unit area of the source dA, per unit unit area of the source dA, per unit frequency dfrequency d, per unit solid angle d, per unit solid angle d in a given direction (in a given direction (θθ,,) (see p. ) (see p. 151-152)151-152)

Units: w mUnits: w m-2-2 Hz Hz-1-1 ster ster-1 -1

dd = da/r = da/r22 dd = = da/rda/r22 = 4 = 4rr22/r/r22 = 4 = 4

Luminosities and magnitudes of stars Luminosities and magnitudes of stars §3.2§3.2

Luminosities and magnitudes of stars Luminosities and magnitudes of stars §3.2§3.2

Luminosity Luminosity is energy passing through closed is energy passing through closed surface encompassing the source (units watts) surface encompassing the source (units watts)

Luminosity L = Luminosity L = IIdAddAddd

If source (star) radiates isotropically, its If source (star) radiates isotropically, its radiation at distance r is evenly distributed radiation at distance r is evenly distributed on a spherical surface of area 4 on a spherical surface of area 4 r r22

Flux Flux is then is then F F = L / 4 = L / 4 r r2 2 (w m(w m-2-2))

F falls off as 1 / rF falls off as 1 / r22

Inverse Square LawInverse Square Law Solar constant is Solar constant is 1365 w m 1365 w m-2-2

Fig 4.3. An energy flux, which, at distance r from a point source, is distributed over an area A, is spread over an area 4A at a distance 2r. Thus, the flux density decreases inversely proportional to the

distance squared.

Brightness, the magnitude scale Brightness, the magnitude scale §4.2-3 §4.2-3 Brightness, the magnitude scale Brightness, the magnitude scale §4.2-3 §4.2-3

In 120 BC, Greek astronomer, Hipparchus, In 120 BC, Greek astronomer, Hipparchus, ranked stars in terms of importance (ie. ranked stars in terms of importance (ie. brightness) brightness) “magnitude” “magnitude”

11stst magnitude were brightest magnitude were brightest 6 6thth magnitude faintest visible stars (later magnitude faintest visible stars (later extended to 0 and -1)extended to 0 and -1)

Without realizing it, Hipparchus based Without realizing it, Hipparchus based his scheme on the sensitivity of the his scheme on the sensitivity of the human eye to flux - logarithmic scale, human eye to flux - logarithmic scale, not a linear one.not a linear one.

Perceived brightness Perceived brightness log (actual flux) log (actual flux)

Rigel & Betelgeuse - 0Rigel & Betelgeuse - 0thth Magnitude Stars Magnitude Stars Rigel & Betelgeuse - 0Rigel & Betelgeuse - 0thth Magnitude Stars Magnitude Stars

Brightness and the magnitude scaleBrightness and the magnitude scale Brightness and the magnitude scaleBrightness and the magnitude scale

Magnitude scale later standardized so that mag. Magnitude scale later standardized so that mag. = 1 is exactly 100 x brighter than mag. = 6= 1 is exactly 100 x brighter than mag. = 6

Difference of 5 mag = factor 100 in brightnessDifference of 5 mag = factor 100 in brightnessDifference of 1 mag = factor 2.512 in Difference of 1 mag = factor 2.512 in brightness i.e. (2.512)brightness i.e. (2.512)55 = 100 = 100

Note:Note: smaller mag is brighter star smaller mag is brighter starWe can quantify this definition of magnitude We can quantify this definition of magnitude scale: Ratio of two brightness (flux) scale: Ratio of two brightness (flux) measurements is related to the corresponding measurements is related to the corresponding magnitudes by bmagnitudes by b11/b/b22 = 100 = 100 (m(m

22-m-m11)/5)/5

bb11 and b and b22 are fluxes and m are fluxes and m11 and m and m22 are are

magnitudesmagnitudes NB that it is bNB that it is b11/b/b22 and m and m22 - m - m11

Brightness and the magnitude scaleBrightness and the magnitude scale Brightness and the magnitude scaleBrightness and the magnitude scale

This is usually expressed in the form:This is usually expressed in the form: mm22 - m - m11 = 2.5 log = 2.5 log1010 (b (b11/b/b22) )

Note that it is mNote that it is m22 - m - m11 on the left and b on the left and b11/b/b22 on the right on the right ratio apparent mag. difference ratio apparent mag. difference         brightness  (b         brightness  (b11/b/b22) ) m m22-m-m11

1 = 101 = 1000 0 0 10 = 1010 = 101 1 2.5 2.5 100 = 10100 = 1022 5.0 5.0 1000 = 101000 = 1033 7.5 7.5 10,000 = 1010,000 = 1044 10.0 10.0 10108 8 20.020.0

Brightness and the magnitude scaleBrightness and the magnitude scale Brightness and the magnitude scaleBrightness and the magnitude scale

1528 Latin translation of Ptolemy’s1528 Latin translation of Ptolemy’s Almagest based on Hipparchus of 120 BC Almagest based on Hipparchus of 120 BC

1528 Latin translation of Ptolemy’s1528 Latin translation of Ptolemy’s Almagest based on Hipparchus of 120 BC Almagest based on Hipparchus of 120 BC

Brightness and the magnitude scaleBrightness and the magnitude scale Brightness and the magnitude scaleBrightness and the magnitude scale

This is usually expressed in the form:This is usually expressed in the form: mm22 - m - m11 = 2.5 log = 2.5 log1010 (b (b11/b/b22) )

Note that it is mNote that it is m22 - m - m11 on the left and b on the left and b11/b/b22 on the right on the right ratio apparent mag. difference ratio apparent mag. difference         brightness  (b         brightness  (b11/b/b22) ) m m22-m-m11

1 = 101 = 1000 0 0 10 = 1010 = 101 1 2.5 2.5 100 = 10100 = 1022 5.0 5.0 1000 = 101000 = 1033 7.5 7.5 10,000 = 1010,000 = 1044 10.0 10.0 10108 8 20.020.0

Brightness and the magnitude scaleBrightness and the magnitude scaleBrightness and the magnitude scaleBrightness and the magnitude scale

Since brightness of a given star depends Since brightness of a given star depends on its distance, we define:on its distance, we define:Apparent magnitudeApparent magnitude, , mm (this represents (this represents flux) = magnitude measured from Earthflux) = magnitude measured from Earth

Absolute magnitudeAbsolute magnitude, , MM (this represents (this represents luminosity) = magnitude that would be luminosity) = magnitude that would be measured from a standard distance of measured from a standard distance of 10 parsecs (chosen arbitrarily)10 parsecs (chosen arbitrarily)

m - M = 2.5logm - M = 2.5log1010 (B/b) (B/b) Where B is the flux measured at 10 pc Where B is the flux measured at 10 pc and b is flux measured at distance d to and b is flux measured at distance d to the starthe star

Brightness and the magnitude scaleBrightness and the magnitude scale Brightness and the magnitude scaleBrightness and the magnitude scale

Using inverse square law, B/b = (d/10 pc)Using inverse square law, B/b = (d/10 pc)2 2

we getwe get

m - M = 2.5 logm - M = 2.5 log1010 (d/10) (d/10)2 2 = 5 log= 5 log1010 (d/10) = 5 (log (d/10) = 5 (log1010 d - d -

loglog1010 10 ) 10 )

The last term is just = 1 so we have The last term is just = 1 so we have m - M = 5 logm - M = 5 log1010 d - 5 or m - M = 5 log d - 5 or m - M = 5 log1010

d/10d/10

m - M is called the distance modulus and m - M is called the distance modulus and will appear often.will appear often.

d is distance to the star in parsecs.d is distance to the star in parsecs.

Simple problemsSimple problemsSimple problemsSimple problems (a) What is the absolute magnitude M of the (a) What is the absolute magnitude M of the Sun?Sun?

(b) How much brighter or fainter in luminosity (b) How much brighter or fainter in luminosity is the star Proxima Centauri compared to the is the star Proxima Centauri compared to the Sun?Sun?

Needed data:Needed data:mmsunsun = -26.7; m = -26.7; mproximaproxima = 11.05 = 11.05Parallax of proxima = 0.77”Parallax of proxima = 0.77”1 pc = 206,265 AU1 pc = 206,265 AU

(c) Total magnitude of a triple star is 0.0. (c) Total magnitude of a triple star is 0.0. Two of its components have magnitudes 1.0 and Two of its components have magnitudes 1.0 and 2.0. What is magnitude of the third component?2.0. What is magnitude of the third component?