4.3: Radio source energy budget.

In document Optical emission lines in radio sources of intermediate power (Page 169-171)

In this section we derive estim ates for th e energy flux in th e radio je t via a consideration of th e overall energy budget of th e source and using th e source ages p resen ted in th e previous section. D eterm ining th e to ta l energy in a radio lobe over a know n tim e scale yields an average pow er for th e delivery of energy via th e jets. T he purpose of this section is to establish the fractio n of th e to ta l energy budget of these radio source which we observe as radio em ission.

In th e following discussion we address only th e class II radio sources. Energy b u d g ets of class I radio sources have been considered in detail by Bicknell (1986). A full tre a tm e n t of th e energy budget for th e class I sources PK S 0344-345 and PK S 1517-283 requires m ore detailed inform ation on jet surface b rig h tn ess and je t opening angle th a n is available from our d a ta. M oreover, we recall from th e previous discussion th a t our velocity analysis is a p p aren tly not p a rtic u la rly valid for th e class I radio sources considered above.

In th e previous section dealing w ith th e velocities an d ages of th e radio lobes we n o ted th a t th e age estim ates are strictly lower lim its on th e tru e age of the radio source. This is because th ere m ay be old p lasm a in th e lobes w hich is u n d etectab le at our p a rtic u la r frequency an d w ith our p a rtic u la r dynam ic range. We note, however th a t this is not a problem for th e energy budget analysis. We derive th e lobe energies from th e d etected p a rts of th e source. T h e velocities and ages are derived from these sam e p a rts of th e source. T h u s th ere is no restrictio n im plied by considering only a section of th e radio source.

T h e to ta l energy of a radio lobe evolves due to th e com bined effect of energy in p u t from th e jet energy losses due to rad iatio n an d work perform ed in expanding

th e lobe. Bicknell (1986) gives th e expression for th e energy budget of a radio lobe:

^ = Fe - £ L,exp - C L (4.9)

w here El is th e to ta l lobe energy, Fe is th e energy flux in to th e lobe, El^ \ p is the

ra te of work done by th e expansion and Cl is th e to ta l syn ch ro tro n plus inverse

C o m p to n lum inosity. We discuss th e lobe energy, th e work done in inflating the lobe a n d th e rad iativ e lum inosity in th e following subsections.

4.3.1:

Total lobe energy.

T h e lobe energy com prises th e in te rn a l energy-density e, th e kinetic energy- d en sity \ p v 2 and th e m agnetic field energy density B 2/ 8 n in te g ra te d over the lobe volum e.

E L = J v { e + l pv2 + V ) d3v( 4 i o )

W e have ignored g ra v ita tio n a l p o te n tia l energy. In applying this expression to class II radio lobes we identify th e plasm a velocity v w ith th e backflow speed of th e p lasm a w ith respect to th e in terg alactic m edium , uy, an d ignore any oth er possible in te rn a l m otions such as turbulence. T h e velocities derived for class II sources in th e previous section on sp ectral aging are th e velocities of the flow w ith resp ect to th e h o tsp o t. Since th e h o tsp o t m ay itself be m oving w ith respect to th e in terg alactic m edium w ith speed Vh, th e velocity we observe is u0bs = v / + Vh-

To sim plify equation (4.10) we assum e th a t th e energy densities do not vary g reatly over th e lobe, hence we m ay replace th e in teg ral by m ultip licatio n w ith th e lobe volume. It is th e n u n d ersto o d th a t all p a ram eters represent m ean energy densities over th e lobe. T his assum ption is m ade for th e o th e r co n trib u tio n s to th e energy budget discussed below.

To estim ate th e lobe volum e we assum e cylindrical geom etry w ith diam eter

Dl an d length i from th e radio core to th e h o tsp o t a t the end of th e lobe (see figure 31). We fu rth e r estim ate th e ra te of change of lobe energy by an average over th e age of th e source: d E L / d t ~ El/t, where r is th e age of th e source. A little m an ip u latio n yields:

d E L

dt

* + ö P vf + lC=

*p i e

4 T (4.11)

T h e backflow velocity w ith respect to th e in terg alactic m edium is designated vj .

T his expression for th e ra te of change of th e to ta l lobe energy includes a c o n trib u tio n from th e rm a l m aterial. Ii is quite clear from studies of F arad ay de­ p o larisatio n (e.g. B u rn 1966) th a t th ere is a negligible am ount of th erm al m aterial

I n t e r g a l a c t i c medium S h o c k e d IGH C o c o o n w i t h b a c k f l o w L ob e d i a m e t e r H o t s p o t

B

f i e l d

B

f i e l d Bow s h o c k L o be l e n g t h J? V o t e =

Figure 4.31:

Diagram of a class II radio lobe defining some of the geometrical

In document Optical emission lines in radio sources of intermediate power (Page 169-171)