4.4
Changing the pressure
p0
Throughout the previous analysis in this chapter and in Chapter3we have keptp0fixed. We turn now to
consider the effect of changing the value of the plasma pressurep0. First we recall D=D0 T3 0 p0L , V =V0 T3 0 p0L , R=R0 p0L T7/2. (4.41)
From expressions (4.41) we see that increasing the value of the pressurep0will decrease the value of the
thermal conduction and compressive viscosity parametersDandV (and hence decreasedandε) but will increase the value of the radiative cooling parameterR(and hence increaser).
4.4.1
Thermal Conduction
For thermal conduction we recall that the period ratio is unity ford= 0and then decreases asdincreases from zero until an optimum value ofdfor which the period ratio has a minimum. After this optimum value ofdthe period ratio tends again to unity. In terms of the pressure then, if an increase in pressure decreases the value ofd, the period ratio will increase if the starting value ofdwas lower than the optimumdvalue and the period ratio will decrease if the starting value ofdwas higher than the optimum value. In Fig.4.11
we display the period ratio as a function of pressurep0under thermal conduction for three cases. In the
first case (solid curve) the loop temperature isT0 = 2MK and the loop half-length isL= 25Mm. From
this we note the behaviour of the period ratio with respect to the pressurep0. As expected, forp0→0and
p0 → ∞the period ratio is unity as this corresponds tod→ ∞andd→0respectively. Between these
two extremes the period ratio is minimised for some value of pressurep0. We can determine the value of
the pressure for which the period ratio is minimised by considering the formula
pmin= (γ−1)κ0T03 dminγL r γR ˜ µ , (4.42)
wheredmin = 0.291 was calculated in Chapter3 to be the value of the thermal conduction parameter
which gives the lowest value of the period ratio. As an example,pmin= 0.029Pa for a loop of half-length
L= 25Mm and temperatureT0= 2MK, where in mks units we useγ= 5/3,R= 8.3×103,µ˜= 0.6and
κ0 = 10−11. In Table4.5we give guide values forpminfor typical loop half-lengthsLand temperatures
T0 under thermal conduction. As expected an increase in the temperature T0 or a decrease in the loop
half-lengthLincreases the value ofpmin. Typical values ofpminfor thermal conduction are of the order
of typical coronal pressures.
In Fig. 4.11the dashed curve shows the behaviour of the period ratio with respect top0for a loop of
temperatureT0= 2MK and loop half-lengthL= 50Mm, i.e. the temperature remains the same as in the
previous case (solid curve) but the loop half-length has been doubled. The dashed curve has been displaced to the left of the solid curve under going a compression, i.e. to achieve the same value of the period ratio when we increase the loop half-lengthLwe must reduce the value ofp0. This is obvious from Eq. (4.41)
which shows that if, for example, we double the value of the loop half lengthLwe must half the value of the pressurep0to keepDand hence the period ratio fixed. Finally, in Fig.4.11the dotted curve shows the
4.4 Changing the pressurep0 95
Table 4.5: A table of values forpminin pascals for thermal conduction for various loop temperaturesT0
and half-lengthsLwithdmin= 0.291,γ= 5/3,R= 8.3×103,µ˜= 0.6,κ0= 10−11in mks units.
T0(MK) L= 25Mm L= 100Mm L= 200Mm
1 3.6e-3 9.1e-4 4.5e-4 2 2.9e-2 7.2e-3 3.6e-3
8 1.9 4.6e-1 2.3e-1
10 3.6 9.1e-1 4.5e-1
behaviour of the period ratio with respect top0for a loop of temperatureT0= 3MK and loop half-length
L = 25Mm, i.e. the loop half length remains the same as for the solid curve but we have increased the temperature. The graph shows a displacement to the right of the solid curve and has been subject to a stretching factor, to achieve the same value of the period ratio as the first casep0must be increased ifT0is
increased. Again this is obvious from Eq. (4.41) which shows that if, for example, we double the value of the temperatureT0we must multiply the pressurep0by23= 8to keepDand hence the period ratio fixed.
Figure 4.11: Plot showing the period ratio as a function of loop pressurep0(in Pa). The solid curve is for
T0= 2MK andL= 25Mm, the dashed curve is forT0= 2MK andL= 50Mm and the dotted curve is
forT0= 3MK andL= 25Mm. Damping is due to thermal conduction only.
4.4.2
Compressive Viscosity
For compressive viscosity we recall that the period ratio is initially unity forε = 0and then decreases indefinitely asεincreases. The coronal parameterεis bounded above by2/πfor which the period ratio
4.4 Changing the pressurep0 96
is effectively zero. After this cutoff value ofεthe period ratio may no longer be formed. In terms of the pressure then, if an increase in pressure decreases the value ofε, the period ratio will increase from zero for a value ofp0which givesε= 2/π; we call this value of the plasma pressurepcutof f, where
pcutof f= 2ν0π r γR ˜ µ T 3 0 3γL ; (4.43)
for values ofp0 < pcutof f the period ratio may not be formed. For example, a loop of half-lengthL= 25
Mm and temperatureT0 = 8MK haspcutof f = 0.039Pa. As the value of the pressurep0is increased
frompcutof f the period ratio increases from zero to unity. Forp0 → ∞(equivalent toε →0) the period
ratio tends to unity. In Table4.6we give guide values for pcutof f for typical loop half-lengthsL and
temperaturesT0under compressive viscosity. As expected an increase in the temperatureT0or a decrease
in the loop half-lengthLincreases the value ofpcutof f. Values ofpcutof f for coronal loop half-lengthsL
and temperaturesT0are typically smaller than0.055Pa, the value for the pressure used throughout Chapter
3and earlier in this chapter.
Table 4.6: A table of values forpcutof fin pascals for compressive viscosity for various loop temperatures
T0and half-lengthsLwithγ= 5/3,R= 8.3×103,µ˜= 0.6,ν0= 10−17in mks units.
T0(MK) L= 25Mm L= 100Mm L= 200Mm
1 7.6e-5 1.9e-5 9.5e-6 2 6.1e-4 1.5e-4 7.6e-5 8 3.9e-3 9.8e-3 4.9e-3 10 7.6e-2 1.9e-2 9.5e-3
In Fig. 4.12we display the period ratio as a function of pressurep0under compressive viscosity for
three cases. The first case (solid curve) is for temperatureT0 = 8MK and loop half-lengthL = 25Mm,
from which we note that the behaviour of the period ratio with respect to pressure. As expected, the period ratio increases from zero forp0greater than the cutoff value which in this case is0.039Pa, asp0→ ∞the
period ratio tends to unity.
In the second case (dashed curve) we fix the temperature (T0 = 8MK) but increase the loop half-
length (L = 50Mm) and show that as for thermal conduction to achieve the same value of the period ratio as the first casep0must be reduced. The value ofpcutof f is also reduced. Equally, in the third case
(dotted curve) we fix the loop half-length (L = 25Mm) but raise the temperature (T0 = 9MK), again
like thermal conduction to achieve the same value of the period ratio as the first casep0must be increased.
The value ofpcutof f is also increased. Consulting Eq. (4.41) this is not surprising, just as for thermal
conduction doubling the loop length may be counteracted by decreasing the pressure by half and doubling the temperature may be counteracted by increasing the pressure eight-fold.
4.4 Changing the pressurep0 97
Figure 4.12: Plot showing the period ratio as a function of loop pressurep0(in Pa). The solid curve is for
T0= 8MK andL= 25Mm, the dashed curve is forT0= 8MK andL= 50Mm and the dotted curve is
forT0= 9MK andL= 25Mm. Damping is due to compressive viscosity only.
4.4.3
Radiative Cooling
For radiative cooling we recall that the period ratio behaves in a similar manner to that seen under the effect of thermal conduction. The period ratio is initially unity forr= 0and then decreases asrincreases from zero until an optimum value ofrfor which the period ratio has a minimum. After this optimum value ofr
the period ratio tends to unity. In terms of the pressure then, if an increase in pressure increases the value ofr, the period ratio will increase if the starting value ofrwas higher than the optimumrvalue and the period ratio will decrease if the starting value ofrwas lower than the optimum value. In Fig. 4.13we display the period ratio as a function of pressurep0under radiative cooling for three cases. In the first case
(solid curve) the loop temperatureT0 = 1MK and the loop half-lengthL = 25Mm. From this we note
the behaviour of the period ratio with respect to the pressurep0. As expected, forp0 →0 andp0 → ∞
the period ratio is unity as this corresponds tor→0andr → ∞respectively. Between these two limits the period ratio is minimised for some value of pressurep0. We can evaluate the value of the pressure for
which the period ratio is minimised by considering the formula
pmin= γ3/2m2 pR5/2T 7/2 0 rmin (γ−1)˜µ5/2χL , (4.44)
wherermin = 1.259 was calculated to be the value of the radiative cooling parameter which gives the
lowest value of the period ratio. As an example, pmin = 1.86Pa for a loop of half-length L = 25
Mm and temperature T0 = 1 MK, where in mks units we useγ = 5/3, R = 8.3×103, µ˜ = 0.6,
4.4 Changing the pressurep0 98
half-lengthsLand temperaturesT0under radiative cooling. As expected an increase in the temperatureT0
or a decrease in the loop half-lengthLincreases the value ofpmin. Typical values ofpminfor radiative
cooling are much higher than typical coronal pressures.
Table 4.7: A table of values forpmin in pascals for radiative cooling for various loop temperaturesT0
and half-lengthsLwithrmin = 1.259,γ = 5/3,R = 8.3×103,µ˜ = 0.6,mp = 1.6726×10−27 and
χ= 5.51×10−30in mks units. T0(MK) L= 25(Mm) L= 100Mm L= 200Mm 1 1.9 4.6e-1 2.3e-1 2 21.0 5.2 2.6 8 2689.8 672.4 336.2 10 5873.5 1468.4 734.2
Figure 4.13: Plot showing the period ratio as a function of loop pressurep0(in Pa). The solid curve is for
T0= 1MK andL= 25Mm, the dashed curve is forT0= 1MK andL= 50Mm and the dotted curve is
forT0= 2MK andL= 25Mm. Damping is due to radiative cooling only.
In the second case (dashed curve) we show the behaviour of the period ratio with respect top0for a loop
of temperatureT0 = 1MK and loop half-lengthL = 50Mm, i.e. the temperature remains the same as
in the previous case (solid curve) but the loop half-length has been doubled. The dashed curve has been displaced to the left and has been compressed in relation to the solid curve, i.e. to achieve the same value of the period ratio when we increase the loop half-lengthLwe must reduce the value ofp0. This is obvious
from Eq. (4.41) which shows that if, for example, we double the value of the loop half lengthLwe must half the value of the pressurep0to keepRand hence the period ratio fixed. In the third case (dotted curve)