9.4  The EM approximations

Ith = Ab(Z)   ó õ T   C(T, Ne)  Ne NH  dh dT dT

(21)

Many authors (e.g. Pottasch 1963; Jordan and Wilson 1971) approximate the above expression by removing an averaged value of C(T) from the integral:

Ith = Ab(Z)   < C(T) >   ó õ h   Ne NH dh

(22)

A suitably defined volume line emission measure EM_L can therefore be defined, for each observed line of intensity I_ob:

EML º  Iob Ab(Z)   < C(T) >      [cm-5]

(23)

The relative abundances of the elements are derived in order to have all the line emission measures of the various ions lie along a common smooth curve.

9.4.1  A different approach. The DEM_L method

A different approach was proposed by Widing and Feldman (1989):

extract from the integral an averaged value of the DEM of the line, that here is termed the line DEM DEM_L:

DEML º < Ne NH  dh dT >     [cm-5 K-1]

(24)

such that for each line of observed intensity I_ob:

DEML º   Iob Ab(Z)   ó õ T   C(T) dT

(25)

A plot of the A_b(Z)   DEM_L = I_ob / ò_T  C(T) dT values displayed at the temperatures T_max is used to deduce relative element abundances, adjusting them in order to have a continuous sequence of the A_b(Z)   DEM_L values.

9.4.2  Various problems

Only when the two lines have similar C(T) and the DEM distribution is relatively flat would one expect that the DEM factors out from the integrals:

ó õ T    C1(T, Ne)  DEM (T)  dT ó õ T    C2(T, Ne)  DEM (T)  dT = ó õ T    C1(T, Ne)  dT ó õ T    C2(T, Ne)  dT

(26)

If the above equality holds, then it is possible to deduce the relative abundances directly from the observed intensities and the contribution functions, because:

Ab(X1) Ab(X2) = I1 ·  ó õ T    C2(T, Ne)  dT I2 ·  ó õ T    C2(T, Ne)  dT =  DEML(X2) DEML(X1)

(27)

i.e. the DEM method and the DEM_L method are equivalent.

However, this is not always the case. The Figures below show that, in the case of the famous Skylab plume observation of Widing and Feldman (1992), the DEM_L approximation overestimated the FIP effect by a factor of 10.

Figure 45: [from Del Zanna et al. 2003] The DEM_L values for the Skylab plume. The Ne and Mg values indicate the need to modify the adopted Ne/Mg photospheric abundance ratio, in order to obtain a smooth distribution of DEM_L values. The DEM(T) derived from the same data is plotted for comparison (dashed line).

Figure 46: [from Del Zanna et al. 2003] The I_ob / (A_b * C(T)) curves for the Skylab plume, assuming photospheric abundances. The data indicate an isothermal distribution at log T=5.9 and are consistent with no FIP effect present. The emission measure EM(0.1) values (see Del Zanna et al., 2002 for the definition), calculated with the DEM of Fig. 46 and a Dlog T=0.1 are also shown (filled circles).

Many other effects can significantly affect results. For example:

- Blending

- Density effects

- The problem with the Na- and Li-like ions

See the many Del Zanna (et al.) papers...

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