Luminosity: Worked Examples and Complete Solutions

(Example)

Luminosity: Worked Examples and Complete Solutions

This companion develops numerical skill with the luminosity relations introduced in the definition article.

Unless otherwise stated, use

L_⊙	= 3.828 × 10²⁶ W,	(1)
R_⊙	= 6.957 × 10⁸ m,	(2)
T_eff,⊙	= 5772 K,	(3)
1 au	= 1.495978707 × 10¹¹ m,	(4)
1 pc	= 3.085677581 × 10¹⁶ m,	(5)
σ	= 5.670374419 × 10⁻⁸ W m⁻² K⁻⁴.	(6)

Part I: Problems

Problem 1: energy radiated by the Sun

Using the nominal solar luminosity, calculate how much energy the Sun radiates in:

one second;
one hour;
one Julian year.

Problem 2: solar flux at one astronomical unit

Treat the Sun as an isotropic emitter.

Use

L F = 4πd2-

(7)

to calculate the solar bolometric flux at one astronomical unit.

Problem 3: infer luminosity from flux and parallax

A star has measured bolometric flux

Fbol = 1.28 × 10− 11 W m −2

(8)

and parallax

ϖ = 20.0 mas.

(9)

Assume the simple high-signal-to-noise inversion

-1000--- d(pc ) = ϖ (mas ).

(10)

Find:

the distance in parsecs;
the luminosity in watts;
the luminosity in solar units.

Problem 4: luminosity from radius and effective temperature

A star has

R	= 2.00R_⊙,	(11)
T_eff	= 6500 K.	(12)

Use

( )2 ( )4 L-- = R--- -Teff- L⊙ R ⊙ Teff,⊙

(13)

to find L∕L_⊙.

Problem 5: infer stellar radius

A star has

L	= 25L_⊙,	(14)
T_eff	= 8000 K.	(15)

Find its radius in solar radii.

Problem 6: equal-luminosity unresolved binary

Two identical stars each have luminosity

L = 1.0L ⊙.

(16)

They are unresolved.

Find:

the total luminosity;
the magnitude difference between the unresolved pair and one component.

Problem 7: unequal unresolved binary

An unresolved binary has

L₁	= 4.0L_⊙,	(17)
L₂	= 1.0L_⊙.	(18)

Find:

the total luminosity;
how many magnitudes brighter the unresolved system is than star 1 alone.

Problem 8: spectral luminosity conversion

At wavelength

λ = 550 nm,

(19)

a source has spectral luminosity

12 − 1 L ν = 2.00 × 10 W Hz .

(20)

Find the corresponding:

L_λ in W m⁻¹;
L_λ in W nm⁻¹.

Problem 9: luminosity uncertainty from flux and distance

A source has

F	= (4.00 ± 0.08) × 10⁻¹² W m⁻²,	(21)
d	= 100.0 ± 2.0 pc.	(22)

Estimate the fractional luminosity uncertainty using first-order independent-error propagation.

Problem 10: luminosity uncertainty from radius and temperature

A star has independently measured

	= 1.0%,	(23)
	= 1.0%.	(24)

Estimate the fractional uncertainty in luminosity inferred from

L ∝ R2T 4.

(25)

Problem 11: bolometric magnitude

A star has

L = 100L ⊙.

(26)

Using

Mbol,⊙ = 4.74,

(27)

find its bolometric absolute magnitude.

Problem 12: a complete binary star luminosity example

A detached binary contains:

R₁	= 1.20R_⊙,	T₁	= 6200 K,	(28)
R₂	= 0.90R_⊙,	T₂	= 5200 K.	(29)

The system is at

d = 40.0 pc.

(30)

Assume both stars radiate according to the effective-temperature relation.

Calculate:

L₁∕L_⊙;
L₂∕L_⊙;
total luminosity;
bolometric flux at Earth;
the magnitude brightening caused by adding star 2 to star 1.

Figure 1. In an unresolved binary, energy outputs add linearly even though astronomical magnitudes combine logarithmically.

Part II: Complete Solutions

Solution 1

Luminosity is energy per unit time:

E = LΔt.

(31)

For one second,

|--------------------| E1s = 3.828 × 1026 J.| ----------------------

(32)

For one hour,

E_1h	= (3.828 × 10²⁶)(3600)	(33)
	= 1.378 × 10³⁰ J.	(34)

A Julian year is

365.25 × 86400 = 31557600 s.

(35)

Therefore

E_1yr	= (3.828 × 10²⁶)(31557600)	(36)
	≈ 1.208 × 10³⁴ J.	(37)

Solution 2

Use

---L-⊙---- F = 4π(1 au )2 .

(38)

Substituting the nominal values,

F	=	(39)
	≈ 1361 W m⁻².	(40)

This is the expected scale of the total solar irradiance near Earth.

Figure 2. Solar luminosity is an intrinsic power, while the solar flux at Earth follows from geometric spreading over a sphere of radius one astronomical unit.

Solution 3

The distance is

d	=	(41)
	= 50.0 pc.	(42)

Convert to meters:

d	= 50.0(3.085677581 × 10¹⁶)	(43)
	= 1.54284 × 10¹⁸ m.	(44)

Then

L	= 4πd²F	(45)
	= 4π(1.54284 × 10¹⁸)²(1.28 × 10⁻¹¹)	(46)
	≈ 3.83 × 10²⁶ W.	(47)

Therefore

|------------| |-L- | |L ⊙ ≈ 1.00. | -------------

(48)

The numbers were deliberately chosen to produce a solar-luminosity example.

Solution 4

Use

	= (2.00)²⁴	(49)
	≈ 4(1.1261)⁴	(50)
	≈ 6.43.	(51)

Thus the star radiates about 6.4 times the solar luminosity.

Figure 3. Stellar luminosity grows quadratically with radius and with the fourth power of effective temperature.

Solution 5

Start with

( )2 ( )4 -L- = -R-- T-- . L ⊙ R ⊙ T⊙

(52)

Solve for radius:

|------------------| | ∘ | |-R--= -L∕L-⊙--.| |R ⊙ (T∕T ⊙)4 | --------------------

(53)

Substitute the values:

	=	(54)
	≈ 2.60.	(55)

Solution 6

Luminosities add:

|-------| Ltot = L1 + L2 = |2.0L⊙. | --------

(56)

The magnitude change is

ΔM	= −2.5 log ₁₀	(57)
	= −2.5 log ₁₀2	(58)
	≈−0.753 mag.	(59)

The unresolved pair is 0.753 magnitudes brighter than one component.

Solution 7

The total luminosity is

|--------------| -Ltot =-5.0L⊙.-|

(60)

Relative to star 1,

ΔM	= −2.5 log ₁₀	(61)
	≈−0.242 mag.	(62)

Thus adding the one-solar-luminosity companion makes the unresolved system about 0.24 magnitudes brighter than the primary alone.

Solution 8

Use

c-- Lλ = λ2L ν.

(63)

With

c	= 2.99792458 × 10⁸ ms⁻¹,	(64)
λ	= 550 × 10⁻⁹ m,	(65)

we obtain

L_λ	= (2.00 × 10¹²)	(66)
	≈ 1.982 × 10³³ W m⁻¹.	(67)

Since

−9 1 nm = 10 m,

(68)

the luminosity per nanometer is

|---------------------------| L λ ≈ 1.982 × 1024 W nm −1. | -----------------------------

(69)

Solution 9

The fractional flux uncertainty is

σF 0.08 -F- = 4.00 = 0.020.

(70)

The fractional distance uncertainty is

σd 2.0 d--= 100.0 = 0.020.

(71)

Therefore

	≈	(72)
	=	(73)
	= 0.0447.	(74)

Thus the luminosity uncertainty is approximately

|------| 4.47%. | --------

(75)

Distance dominates even though the fractional distance and flux errors are equal.

Figure 4. Distance uncertainty enters the luminosity relation with a factor of two, while temperature uncertainty enters stellar luminosity with a factor of four.