Gas-Cell Simulator |  Atmospheric Paths |  My Spectra |  Line List Browser |  Blackbody Calculator |  Atmosphere Browser |  Solar Calculator |  Unit Converter
Appendix B: The Doppler Effect

Blackbody Calculator

Appendix B: The Doppler Effect

(click on equations to view enlarged)

The observed frequency ν of light emitted from a source moving with velocity u as depicted in Fig. B1 is given by [4]

,                                     (B1)

where ν is the frequency of the light in the rest frame of the source, and c is the speed of light, 299,792  km/s.

 Fig B1Geometry of a moving source.  If the source is receding, the measured frequency will be decreased (“red shifted”), compared to the frequency in the rest frame of the source.  If the source is approaching, the measured frequency will be increased (“blue shifted”).  Eq. B1 gives the relationship between the frequencies in the two reference frames.

If the velocity is purely radial, then Eq. B1 reduces to

,                                             (B2)

where β2 = (c-u)/(c+u).  Note that in Eq. B2 we have adopted the convention that u > 0 indicates a receding source. For our application, we assume the velocity is purely radial (θ= 0° or 180°) and use Eq. 2.[5]  The magnitude of the Doppler effect for some typical situations is given in Table B1.

 Table B1Examples of Doppler shifts for 1000 cm-1 (10 μm) light source recession velocity (km/s) shift (cm-1) geostationary satellite 3 0.01 low Earth orbit satellite 7 0.02 orbital speed of Earth 30 0.1 typical star 300 1 most distant galaxy 75,000 225

Eq. B2 gives the Lorentz transformation for a monochromatic frequency ν.  However, for a continuous spectrum, we cannot simply scale all frequencies.[6]  We must also apply the Lorentz transformation to the (necessarily finite) aperture collecting the radiation.  An aperture subtending solid angle Ω in the rest frame of the source will appear to have solid angle

(B3)

when receding with velocity u.  Suppose now that the source has a rest-frame radiance of LP(ν).  An aperture of size Ω  in its rest frame will receive a photon flux of

N(ν) = ΩLP(ν)

If the aperture is receding with velocity u, the photon flux received from the receding source will be

If the source is a blackbody at temperature T,  (Eq. 4), we have

If we interpret this radiation as coming from a stationary blackbody, that is, L'(ν) = N'(ν)/Ω', then the effective temperature is

,                                                  (B4)

Thus the spectrum of a receding blackbody appears identical to a cooler, stationary blackbody.

[4] J. D. Jackson, Classical Electrodynamics, 2nd ed., pp 522

[5] The tangential effect is small in most realistic cases anyway.  For example, 1000 cm-1 light (10 mm) from a source moving at 100 km/s perpendicular to the line of sight is shifted only 56×10-6 cm-1.

[6] T. P. Gill, “The Doppler Effect”, Logos Press, Inc., 1965

 Calculation of a Blackbody Radiance Units of Frequency Units of Wavelength Units of Wavenumbers Radiance: Integrating the Planck Equation In-band Radiance: Integrating the Planck Equation over a Finite Range Appendix A: Algorithms for Computing In-band Radiance Appendix B: The Doppler Effect Appendix C: Summary of Formulas References Blackbody Calculator Print Version

GATS Inc.
 Gas-cell Simulator
Microsoft Internet Explorer 6.0 and above
Netscape 6.2 and above
Konqueror 3.0 and above
Safari 2.0 and above
Opera 8.5 and above
Firefox 1.5 and above
Other browsers may work but are not fully tested or supported.
CLOSE  X
Transmittance: ratio of received radiation intensity, I, to incident light intensity, I0

Transmittance: ratio of received radiation intensity, I, to incident light intensity, I0

The information provided will not be shared, sold or used in any way other than to contact users to announce new features.
CLOSE  X

light with wavenumber between σ and σ + dσ

light with wavenumber between σ and σ + dσ
Isotopes are forms of an element whose nuclei have the same atomic number, the number of protons in the nucleus,but different atomic masses because they contain different numbers of neutrons.
CLOSE  X
Cell: model the transmission/radiance of a gas cell. Specify it's length, temperature and pressure, and the vmrs of the absorbing gases.
CLOSE  X

Wavenumber cm-1: the number of
wavelengths of light per centimeter

LINEPAK: The GATS spectral radiance and transmission software library. Performs detailed and accurate line-by-line modeling of molecular absorption. Efficient and flexible, LINEPAK is at the heart of analysis systems for many major atmospheric remote sensing missions, including HALOE, SABER, LIMS, SOFIE, CRISTA, and CLAES.
CLOSE  X
Tangent Path: Model the transmission or radiance of a ray that passes completely through the Earth's atmosphere but does not intersect the Earth. The path is specified by the tangent height, the height at the point of closest approach to the surface. The pressure, temperature and vmrs of absorbing gases at each altitude are chosen from a database of atmospheric states.
CLOSE  X
Slant path: Model the transmission or radiance of a ray between two arbitrary points in the Earth's atmosphere. The points are specified by their heights and the zenith angle from one to the other.
CLOSE  X
VMR: volume mixing ratio. The fractional number of molecules of a species in a volume.

Individual vmrs and their sum must be between 0 and 1.

If the vmrs sum to less than 1, the rest of the gas in the cell is assumed transparent.(Lineshapes for molecules with vmr less than 1 are air-broadened.)
Clicking this will display the data as text in a new browser window. Right-clicking will download the data file to your computer (recommended). These files can be extremely large depending on the spectrum simulated.
Clicking this will open a new browser window suitable for printing.
You have exceeded your daily limit.

To help ensure the availability of our servers, public use is limited to 50 calculations per day. Subscribe now for uninterrupted service. Subscribers also have access to advanced features such as large wavebands, multiple gases, choice of units, radiance spectra, logo-free high-resolution graphics, ascii data files,
full tech support and much more.

Spectroscopy and remote sensing tools for researchers, teachers, and students

Get priority use of advanced, state-of-the-art radiative transfer algorithms--the same ones used by NASA for many remote sensing missions. Subscribers gain access to large wavebands, multiple gases and cells, choice of units, radiance spectra, logo-free high-resolution graphics, data files, full tech support, and much more.

CLOSE  X
Temperature Offset: The model atmosphere (US_Standard, Tropical, etc.) determines the temperature, pressure and gas concentrations at each height in the atmosphere. To adjust the temperature from the model value, enter a temperature offset (from -50 to 50 K). The Atmosphere Browser tool displays the temperature profiles for the model atmospheres.
CLOSE  X
Atmosphere: An atmosphere contains profiles of temperature and gas concentrations at all altitudes. There are six system-supplied atmospheres for Earth and one for Mars. Custom atmospheres can be uploaded from the Atmosphere Browser.
CLOSE  X
Scale Factor for Gas Concentrations: The model atmosphere (US_Standard, Tropical, etc.) determines the gas concentrations at each altitude. To adjust a gas concentration, choose a scale factor, from 0 to 1000. For example, to simulate an atmosphere with 20% more water vapor than the model, enter a scale factor of 1.2 for H2O. Note: while the model atmospheres are physically realistic, using large scale factors can produce unphysical situations where the gas abundance exceeds 100%. If this occurs, an error message will be displayed.
CLOSE  X
The atmosphere model (US_Standard, Tropical, etc.) determines the temperature, pressure and gas concentrations at each height in the atmosphere. To adjust a gas concentration, choose a scale factor other than 1 (from 0 to 1000). For example, to simulate a path with 20% more water vapor, use a scale factor of 1.2 for H2O. The Atmosphere Browser tool displays the temperature, pressure and gas mixing ratios for the model atmospheres..
Below is a list of valid characters:
A-Z      a-z      0-9   _      -      .

(      )       [      ]       :       ;       ,

Create Atmosphere
Modify Atmosphere
Delete Atmosphere
Plot/Extract Species
Atmosphere List