• Flux
  • Luminosity
  • Distance
• The Doppler Effect

Luminosity and Flux

Here we review the definitions of some important quantities that we will use often in the course.

Luminosity, L, is the total energy radiated from an object per second.

• Measured in Watts

Energy flux is the flow of energy out of a surface.

• Measured in Watts/m²

The observed flux (apparent brightness) of an object is the power we receive from it.

• Depends on the distance to the object.
• Measured in W/m²

We often just use the term flux without the "energy" or "observed" qualifier. The unspecified qualifier is determined by the context.

How is the Observed Flux determined?

Make a sphere of radius, r, around an object which is radiating power.

• Such as the sun or a light bulb

All energy radiated from the object must pass through this sphere

• The size of the sphere does not matter!

However, the flow of energy per m² passing through the sphere decreases as the size of the sphere increases.

The flux of energy through the sphere is

r = radius of sphere

L = luminosity of the object

This formula is called the Inverse Square Law because of the dependence on the radius of the sphere.

The radius of the sphere is just the distance to the object.

The flux is what we measure.

We use a telescope (or our eye) and measure a small fraction of the light passing through this sphere.

An Illuminating Example?

• about 1/5 of power goes into light

Confused About Flux?

To reiterate, here are the definitions and usage of energy and observed flux.

Energy flux: F = sigma*T⁴ (W/m²)

• Energy flow out of the surface of a star (or any object).

Observed flux: f = L/(4*r²) (W/m²)

• Apparent brightness
• Energy flow through a sphere of radius r due to a star (or any object) of luminosity L.
• Inverse square law behavior

Observed and Energy Flux are Related

The luminosity of a star is L = 4pR² sT⁴

The total power is energy flux times area.

So for a star with radius R and temperature T.

For example, doubling the distance decreases brightness by factor of 4

You should know that luminosity scales as R²T⁴ and be able to use this information

For example, doubling size increases luminosity by factor of 4, or

doubling temperature increases luminosity by factor of 16

e.g. f_sun = 1 kW/m²

m_A - m_B = 2.5 log (f_B / f_A)

Sample magnitude example

Suppose we have two stars with flux ration f_B/ f_A= 10. What is the difference in their magnitudes.

Using

m_A- m_B = 2.5 log (f_B / f_A)

We have m_A - m_B = 2.5 log (10)

so that m_A - m_B = 2.5

Converting between f and m

We can also write our relation between magnitudes and fluxes as

Flux example, if m_A = 5 and m_B = 0, then using this equation we have f_B / f_A = 10^(5-0)/2.5 so that f_B / f_A = 10² = 100.

Measuring Distance!

If we know the luminosity of an object (such as a star) and measure the flux --

we can determine its distance!

Objects with known luminosity are called standard candles in astronomy.

They are of fundamental importance.

Astronomers use standard candles to measuring distances.

There are very few standard candles and it is a problem to calibrate them (determine L).

The Doppler Effect is a change in the observed frequency of light due to relative motion. Only the motion along the line-of-sight matters.

Water Wave Illustration

Consider Suppose you are traveling a boat and encounter a "water wave" generated by the wake of another boat. The speed at which you transverse the wave will depond upon whether you are traveling with or againt the wave.

• When the boat is traveling into the waves, the peaks hit the bow more rapidly than if the boat were standing still.

• Likewise, when the boat is traveling away from the waves, the peaks hit the rear less rapidly than if the boat were standing still.

Sound wave exhibit the Dopper effect.

• The frequency of sound waves increases as a source approaches the observer, and decreases as it recedes.

You may have noticed this happens when a car or train passes by you. There is a change in pitch between the car is approaching and the car receding.

Electromagnetic Waves - Light

• The Doppler effect also modifies light (photons).
  
• Because atoms emit light at discrete frequencies, we can detect their motion (velocity) by a "shift" in frequency from the expect one.

Astronomers use the shortcut terms blueshift and redshift to refer to the direction (or sign) of the Doppler shift.

Approaching sources

• Spectral lines shifted to higher frequencies
• => short wavelengths.
• Spectrum is blueshifted

Receding sources

• Lines move to lower frequencies
• => longer wavelength
• Spectrum is redshifted

The change in wavelength is proportional to the velocity.

where v_r is the radial velocity

• positive velocity => receding
• negative velocity => approaching

Importance of Doppler Effect

The Doppler effect is very important because it is the only way of measuring the motions of distant objects.

As we shall see later, the Doppler effect allowed Edwin Hubble to deduce that the universe was expanding, and serves as a means to find the distances to distant galaxies.