PDS_VERSION_ID        = PDS3
RECORD_TYPE           = STREAM

OBJECT                = TEXT
   INTERCHANGE_FORMAT = ASCII
   PUBLICATION_DATE   = 2019-06-01
   NOTE               = "Description for Venus Climate Orbiter (VCO,
                         also known as PLANET-C and AKATSUKI) RS data
                         calculation including calibration"
END_OBJECT            = TEXT

END


calculation to derive the current product
=========================================

  This file is created by extracting texts from [IMAMURAETAL2011].


  Schematic diagram of radio occultation experiments
  --------------------------------------------------

    The following two diagrams, global view and close-up view, show the
    geometric relation among the Earth, Venus, Akatsuki, Venusian Atmosphere,
    and ray path of the transmitted wave using Ultra-Stable Oscillator from
    Akatsuki. Note that the bending of the ray path for each is exaggerated.


      global view
      -----------

                   (2) The wave propagates through the Venusian atmosphere
                       and the ray path of the wave is bended.
                         ..------------..
                      .''    _______     ``.
                     /   ..''       ``..    \
                    / .''  ray path     ``.  \
                   |.'       _______       `.|
                  .'        /       \        `.
                .' |       |         |       | `.   .__
              .'   |       |  Venus  |       |   `. |`. (1) transmission
            .'     |       |         |       |     `.  `.    of the wave
         Earth     |        \_______/        |       `.  `.
                    \                       /          `.  `.
      (3) The wave   \                     /           Akatsuki
          is received `.                 .'
          by ground     ``-------------''
          station at the Earth.


      close-up view
      -------------

                                    asymptotes
                                  `.           .'
                                    `.       .'
                                      `.   .'\
                                        `x'  | ``alpha''
                                       .' `. /
                              .------.'-----`.--------.
                           .-'     .'         `.       `-. Venus' atmosphere
                        .-'      .'             `.        `-.
                     .-'       .'                 `.         `-.
                    /        .'__..------+------..__`.          \
                  .'       ..'' ray path |          ``..         `.
                 /       .'              |              `.   .__
                |      .'\               | ``r0''        /`. |`. transmission
                     .'   \    ..--------|-------..     /   `.  `. of the wave
                   .'      \  ' Venus    |         `   /      `.  `.
        ray path .'         \     body   |            /         `.  `.
               .'            \           |           /   ray path `.  `.
             .'               \          |          /               `.
           .'                  \         |         /               to Akatsuki
         .'               ``a'' \        |        / ``a''
        to Earth                 \       |       /
                                  \      |      /
                                   \     |     /
                                    \    |    /
                                     \   |   /
                                      \  |  /
                                       \ | /
                                        \|/
                                         +
                                    center of Venus


    Note that the variables -- that are enclosed with `` '' -- are described
    in the next section.


  Variables, equations, and the parameters used
  ---------------------------------------------

    -- variables --

      name   label name if any      description
      -----  ---------------------  ------------------------------------------
      r_0    RADIUS (L3)            distance of the ray from the center of
                                    Venus at closest approach to Venus
      alpha  BENDING_ANGLE (L3)     asymptotic bending angle for a radially
                                    symmetric atmosphere.
      a      IMPACT_PARAMETER (L3)  ray impact parameter
      n      REFRACTIVE_INDEX (L3)  refractive index
      mu     REFRACTIVITY (L3)      refractivity


    -- equations --

        n(r0) * r0 = a                        (Bouguer's rule)   eq. (1)

      where r0 is function of a.


                      /inf  1  dn           1
        alpha   = -2a |    --- -- ------------------------- dr   eq. (2)
                      /r_0  n  dr sqrt( (n * r)**2 - a**2 )

      where dn/dr is partial derivative.


                    1   /inf    [  a    / ( a )        \        ] dalpha
        ln n(r) = - --  |    ln [ --- + | (---)**2 - 1 |**(1/2) ] ------ da
                    PI  /a_1    [ a_1   \ (a_1)        /        ]   da
                                                                       eq. (3)

      where a_1 is function of r. ``alpha'' and ``a'' are calculated from
      the measured atmospheric Doppler shift and the reconstructed velocity
      and position vectors of the spacecraft and the ground stations.

      The refractivity mu is related to the refractive index by

        mu = (n-1) x 10**6                                       eq. (4)

      and mu is the sum of the contributions from the neutral atmosphere and
      the ionosphere:

                       beta * N_e
        mu = V * N  -  ---------- * 10**6                        eq. (5)
                          f**2

      where V is the refractive volume, N is the number density of the
      neutral atmosphere, beta = e**2/(8*pi**2*epsilon*m_e) ~ 40.3 m**3 s**-2
      with e, epsilon and m_e being the elementary charge, dielectric constant
      in vacuum and electron mass, respectively, c is the speed of light, N_e
      is the number density of electrons, and f is the frequency of the
      carrier signal. On the assumption that the neutral atmosphere is well
      mixed, we adopt a constant V of 1.811 x 10**-23 m**3 based on the
      composition of 96.5% CO2 and 3.5% N2 using the constants given in
      [FJELDBO&ESHLEMAN1968]. The neutral and ionospheric contributions are
      separated in altitude (see results of [PAETZOLDETAL2007]). This enables
      us to retrieve the vertical profiles of the neutral atmospheric density
      N and the electron density N_e separately.

      The vertical profile of the neutral atmospheric pressure, p(r), is
      derived from the density profile N(r) by integrating the equation of
      hydrostatic equilibrium:

                            / r_top
        p(r) = p(r_top) + m |        N(r') * g(r') dr'         eq. (6)
                            / r

      where m is the mean molecular mass, g is the acceleration due to
      gravity, and r_top is the adopted upper boundary. The ideal gas law
      relates p(r_top) to the atmospheric temperature T at this height as

        p(r_top) = N(r_top) * k * T(r_top),                    eq. (7)

      where k is the Boltzmann's constant. The boundary condition required
      for the integration is the value T(r_top). The temperature profile
      T(r) is calculated from N(r) and p(r) using the ideal gas law. The
      influence of the adopted T(r_top) on the calculated T(r) is almost
      negligible except at the uppermost 10 km [PAETZOLDETAL2007,
      TELLMANNETAL2009].

      The received signal power varies during each occultation experiment
      due to atmospheric defocusing and absorption. The former must be
      taken into account when retrieving the vertical distribution of
      absorbing materials from the observed signal power.

      Atmospheric defocusing is caused by the radial gradient of the
      refractive index. The defocusing loss is estimated by [ESHLEMAN1973]
                           
        L = ( cos alpha - D (d alpha / d a) )**-1              eq. (8)

      where D is the distance from the spacecraft to the crossing of the
      ray asymptotes.

      Once the defocusing loss is estimated as a function of time from
      trajectory information and the relationship between alpha and a from
      the Doppler measurements, one may calculate the total atmospheric
      absorption from the time series of the received signal power. This
      time series is converted to a height profile of the absorptivity with
      the aid of ray tracing on the assumption of a spherically symmetric
      atmosphere [JENKINS&STEFFES1991, JENKINSETAL1994]. The absorption due
      to CO2, N2, and SO2 is removed from the absorptivity profile based on
      its dependence on the ambient pressure and temperature. The remaining
      absorption will be principally due to H2SO4 vapor.


    -- Summary of the assumed parameters --

      Venus radii: 6052 km
      GM:          3.24858592 x 10**5 km**3 s**-2
      V:           1.81 x 10**-17 m**3


