Application Center - Maplesoft
  Submit your work
Application Center Applications Luminosity Preview

App Preview:

Luminosity

You can switch back to the summary page by clicking here.

Learn about Maple
Download Application



Luminosity

Problem

(A) Find the number which, when raised to the fifth power, equals 100. Then derive the formula relating magnitudes to fluxes. (B) Find the absolute magnitude of Sirius and the Sun. (C) Calculate the bolometric magnitude of Antares. Calculate its radius, given the corrected bolometric magnitude. Calculate the temperature of Rigil Kent. (D) Find the bolometric correction of Sirius.

 

 

Hints:

For problem (B), use log to the base 10.

 

 

Data

 

Apparent Magnitude of Sirius

m := -1.44

-1.44

(2.1)

Bolometric Magnitude of Sirius

bm := -1.53

-1.53

(2.2)

Distance to Sirius in Parsecs

d := 2.64

2.64

(2.3)

Solar Apparent Magnitude

ms := -26.8

-26.8

(2.4)

Distance to Sun in Parsecs

ds := 1/206265

1/206265

(2.5)

Radius of Antares in Units of Solar Radius

ra := 800

800

(2.6)

Temperature of Antares in Kelvins

ta := 3500

3500

(2.7)

Corrected Bolometric Magnitude of Antares

ca := 11.2

11.2

(2.8)

Bolometric Magnitude of Rigil Kent

br := 4.35

4.35

(2.9)

Radius of Rigil Kent in Units of Solar Radius

rr := 1.227

1.227

(2.10)

 

 

Useful Equations

 

 "(log)[10](100)*x= "increment   (x is the number which, when multiplied by the log of 100, gives the increment.)

 

Absolute Magnitude from Apparent Magnitude M and Distance in PC, d

 "abmag(m,d):= m+5-2.5*log[10](d^2)"

 

Bolometric Magnitude

bolmag := proc (radius, temp) options operator, arrow; 4.8+40.3823+(-1)*1.085736204758*ln(4*Pi*radius^2*temp^4) end proc 

 

Bolometric Radius

radius := proc (bolmag, temp) options operator, arrow; .2820947918*sqrt(exp((-1)*.9210340372*bolmag+41.61443618))/temp^2 end proc 

 

Bolometric Temperature

temp := proc (bolmag, radius) options operator, arrow; 17515.70144*((exp((-1)*.9210340372*bolmag)*radius^2)^(1/4))/radius end proc

 

Bolometric Correction

bolcor := M[bol]-m[v]   

 

 

 

Solution (A) Apparent Magnitude, m

A range of five orders of magnitude equals a difference in luminosity of 100. Find the number which, when raised to the fifth power, equals 100.

``

base := solve(100. = logbase^5., {logbase})

{logbase = 2.511886432}, {logbase = .7762155953+2.388945959*I}, {logbase = -2.032158811+1.476449800*I}, {logbase = -2.032158811-1.476449800*I}, {logbase = .7762155953-2.388945959*I}

(4.1)

NULL

logbase = 2.51189

logbase = 2.51189

(4.2)

``

For base 10,

``

log[10](100)*x = 5

2*x = 5

(4.3)

"(->)"

x = 5/2

(4.4)

"(->)"

x = 2.5000

(4.5)

NULL``

Therefore, the difference between two apparent magnitudes (m,n) is related to their fluxes (Fm, Fn) by

 

m-n = -2.5*log[10](F[m]/F[n])

 

 

Solution (B) Absolute Magnitude, M

Absolute magnitude is equal to the apparent magnitude a star would have at a distance of ten parsecs.

NULL

"abmag(m,d):= m+5-2.5*log[10](d^2)"

proc (m, d) options operator, arrow; m+5+(-1)*2.5*log[10](d^2) end proc

(5.1)

``

To find the absolute magnitude of Sirius (apparent magnitude = -1.44; distance = 2.64 pc):

 

abmag(-1.44, 2.64)

1.451980366

(5.2)

"(->)"

1.4520

(5.3)

``

The Sun's apparent magnitude is -26.8 and it is 1/206265 parsec from Earth.

 

abmag(-26.75, 1/206265)

-21.75+5.0*ln(206265)/ln(10)

(5.4)

"(->)"

4.822

(5.5)

``

The Sun's absolute magnitude is 4.8.

 

 

Solution (C) Bolometric Magnitude

The magnitude calculated over all wavelengths is the bolometric magnitude. The formula for the absolute bolometric magnitude in terms of the solar radius is

 

bolmag := proc (radius, temp) options operator, arrow; 4.8+40.3823+(-1)*1.085736204758*ln(4*Pi*radius^2*temp^4) end proc

 

Example: Calculate the absolute bolometric magnitude for Antares, a star with 800 times the solar radius and a temperature of 3,500 K.

 

evalf(bolmag(800, 3500))

-7.52185505

(6.1)

The absolute bolometric magnitude of Antares is given here as -7.5. (Antares varies in luminosity, and its absolute bolometric magnitude is usually taken to be about -7.2.)

 

Example: Calculate the radius of Antares, given the bolometric magnitude (-7.2) and the temperature (3500).

``

 

  radius := proc (bolmag, temp) options operator, arrow; .2820947918*sqrt(exp((-1)*.9210340372*bolmag+41.61443618))/temp^2 end proc

evalf(radius(-7.2, 3500))

689.7933132

(6.2)

NULL

This is an under-estimate. The radius of Antares is actually about 800 times the radius of the Sun. The outer layers of the star are being blown off, which makes calculating the radius difficult.

 

Example: Calculate the temperature of Rigil Kent (Alpha Centauri A), given the bolometric magnitude (4.35) and the radius (1.227 times the solar radius).

 

temp := proc (bolmag, radius) options operator, arrow; 17515.70144*((exp((-1)*.9210340372*bolmag)*radius^2)^(1/4))/radius end proc

evalf(temp(4.35, 1.227))

5807.714415

(6.3)

This is a slight overestimate. The temperature of Rigil Kent is generally taken to be 5,790 K.

 

 

Solution (D) Bolometric Correction

The bolometric correction is a number added to the apparent magnitude to compensate for discrepancies caused by significant amounts of luminosity outside the visible range. Therefore, the correction is largest for stars that radiate primarily in the infrared or ultraviolet.

 

bolcor := M[bol]-m[v]

M[bol]-(-1.44)[v]

(7.1)

Example: Sirius

 

Apparent magnitudes:

 

m = -1.44

-1.44 = -1.44

(7.2)

m[bol] = -1.53

(-1.44)[bol] = -1.53

(7.3)

bolcor = -1.53+1.44

M[bol]-(-1.44)[v] = -0.9e-1

(7.4)

 

 

 

 

 

 

NULL