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Hydrodynamical Scaling Laws for Astrophysical Jets

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Hydrodynamical Scaling Laws for Astrophysical Jets 22nd Texas Symposium on Relativistic Astrophysics at Stanford University, Dec. 13-17, 2004 Hydrodynamical Scaling Laws for Astrophysical Jets Mart´ın Huarte Espinosa ([email protected]) Sergio Mendoza ([email protected]) Instituto de Astronom´ıa, Universidad Nacional Autonoma de Mexico, AP 70-264, Ciudad Universitaria, Distrito Federal CP 04510, Mexico The idea of a unified model for all astrophysical jets has been considered for some time now. We present here some hydrodynamical scaling laws relevant for all type of astrophysical jets, analogous to those of Sams et al. [1996]. We use Buckingham’s Π theorem of dimensional analysis to obtain a family of dimensional relations among the physical quantities associated with the jets. 1. Introduction Although the first report of an astrophysical jet was made by Curtis [1918], these objects were extensively studied much later with radio astronomy techniques [Reber 1940]. Quasars, and radiogalaxies were discov- ered and later gathered in a unified model which pro- posed a dusty torus around the nucleus of the source [Antonucci and Miller 1985]. Years later, some galac- tic sources showed similar features to the ones pre- sented by quasars and radiogalaxies, i.e. relativistic fluxes, a central engine, symmetrical collimated jets, radiating lobes, and apparent superluminal motions [Sunyaev et al. 1991]. Optical and X-ray observa- tions showed other similar non–relativistic sources in the galaxy associated to H-H objects [Gouveia Dal Pino 2004]. Lately the strong explosions found in long Gamma Ray Bursts, had been modelled as collapsars, Figure 1: Astrophysical jets are very common and exist in which a jet is associated to the observed phenom- 5 in many different sizes. On the left, extending ∼ 10 pc, ena, in order to explain the observations [Kulkarni FR I and FR 2 sources are shown. The upper right panel et al. 1999, Castro-Tirado et al. 1999]. shows the micro–quasar SS 433. It presents relativistic The similarities between all astrophysical jets, fluxes and apparent superluminal motions analogous to mainly those between quasars and micro–quasars, and those in quasars. The lower right panel shows jets the scaling laws for black holes proposed by Sams et al. associated to Herbig–Haro objects with lengths −1 [1996] and Rees [1998] made us search for the possi- ∼ 10 − 10 pc. All jets have a condensed (sometimes ble existence of some hydrodynamical scaling laws for compact) object accreting matter from their astrophysical jets. surroundings. There is also an accretion disc around the The present work presents a few mathematical re- central condensed object and a pair of symmetrical collimated jets that end up in radiating lobes. The lations that naturally appear as a consequence of di- images were taken from Bridle [1998], Paragi et al. [2001] mensional analysis and Buckingham’s Π theorem. We and and hubblesite.org. begin by considering some of the most natural phys- ical dimensional quantities that have to be included in order to describe some of the physical phenomena related to all classes of jets. With this and the use there are some essential physical ingredients that must of dimensional analysis we then calculate the dimen- enter into the description of the problem. To begin M sional relations associated to these quantities. Finally, with, the mass of the central object must accrete we briefly discuss these relations and their physical material from its surroundings at an accretion rate M˙ B relevance to astrophysical jets. Now, because gravity and magnetic fields are necessary in order to generate jets, Newton’s constant of gravity G and the velocity of light c must be taken into account. If in addition there is some characteris- 2. Analysis tic length l (e.g. the jet’s length), a characteristic den- sity ρ (e.g. the density of the surrounding medium) A complete description fortheformationofanas- and a characteristic velocity v (e.g. the jet’s ejection trophysical jet is certainly complicated. However, velocity), then the jet’s luminosity (or power) L is a 1607 1 22nd Texas Symposium on Relativistic Astrophysics at Stanford University, Dec. 13-17, 2004 function related to all these quantities in the following et al. 1998, Camenzind 1999, Ford and Tsvetanov manner 1999, Vilhu 2002, Wu et al. 2002, Cherepashchuk et al. 2003, Smirnov et al. 2003, Calvet et al. 2004, Mendoza et al. 2004, Mirabel 2004]. ˙ L = L(M, M, c, G, B, l, v, ρ). (1) ¿From equation (2) it is found that Using Buckingham’s Π theorem of dimensional analy- √ sis [Buckingham 1914, Sedov 1993] the following non– GM 3/2 Mc2 3/2 . trivial dimensionless parameters are found Π6 := Π2 Π3 = c2 B (9) L GM˙ Bc1/2M This relation defines a length rj given by Π1 = , Π2 = , Π3 = , (2) Mc˙ 2 c3 M˙ 3/2 lM˙ ρc3M 2 1/3 2/3 1/3 −2/3 M c 2 M B 4 , 5 . r ∝ ≈ . Π = Π = 3 j 2/3 10 pc (10) Mc M˙ B M 1G From the parameter Π2 it follows that For typical extragalactic radio sources and µ– quasars it follows from equations (7) and (8) that 4 rj ∝ 10 pc and rj ∝ 10 pc respectively. These lengths GM M˙ 1 . are fairly similar to the associated length of these jets. Π2 = c2 M c In other words, if we identify the length rj as the length of the jet, then a constant of proportionality Since the quantity ∼ 1 is needed in equation (10), and so M τ ≡ , (4) M 1/3 B −2/3 M˙ rj ≈ 100 pc. (11) M 1G defines a characteristic time in which the central ob- ject doubles its mass, then using equation (4) we can Since equation (9) is roughly the ratio of the write Π2 as Schwarzschild radius rS to the jet’s length rj,then Π6 << 1, i.e. rs Π2 = , (5) 2τc 3/2 Bl3/2 GM 2/l r Π6 = << 1, (12) where S is the Schwarzschild radius. This relation (Mc2)2 naturally defines a length which in turn implies that λ ∼ cτ, (6) c4 B<< ≈ 23 M/ −1 . G3/2M 10 ( M) G (13) which can be thought of as the maximum possible τ length a jet could have, since is roughly an upper The right hand side of this inequality is the max- limit to the lifetime of the source. imum upper limit for the magnetic field associated In what follows we will use the following typical to the accretion disc around the central object. For values “extreme” micro–quasars like SS 433 and GRB’s the magnetic field B 1016 G, so that this upper limit 8−9 −1 works better for those objects [Meier 2002, Trimble M ≈ 10 M,B≈ 100 G, M˙ ≈ 1M yr , and Aschwanden 2004]. 7−10 4−5 L ≈ 10 L,rj ≈ 10 pc, (7) From equation (2) it follows that and LM˙ ≡ Π1 , 0−1 −(8−6) −1 Π7 2 = 2 2 M ≈ 10 M,B∼ 100 G, M˙ ≈ 10 Myr , Π2Π3 B M G L ≈ 2−4 L ,r≈ 0−1 , 10 j 10 pc (8) and so for quasars and µ–quasars respectively [Meier 2002, Trimble and Aschwanden 2004, Blandford 1990, Car- −1 B 2 M 2 M˙ illi et al. 1996, Lovelace and Romanova 1996, Rob- L ∝ −7 L . 10 −1 (15) son 1996, Reipurth et al. 1997, Ferrari 1998, Koide 1G M Myr 1607 2 22nd Texas Symposium on Relativistic Astrophysics at Stanford University, Dec. 13-17, 2004 For the case of quasars and µ–quasars, using the E. M. Gouveia Dal Pino, http://www.arxiv.org/ typical values of equations (7) and (8) it follows that astro-ph/0406319 (2004). 15 8 the power L ∝ 10 L and L ∝ 10 L respectively. S. R. Kulkarni, S. G. Djorgovski, S. C. Odewahn, J. S. In order to normalise it to the observed values, we Bloom, R. R. Gal, C. D. Koresko, F. A. Harrison, can set a constant of proportionality ∼ 10−6 in equa- L. M. Lubin, L. Armus, R. Sari, et al., 398, 389 tion (15). With this, the jet power relation is given (1999). by A. J. Castro-Tirado, M. R. Zapatero-Osorio, N. Caon, L. M. Cairos, J. Hjorth, H. Pedersen, M. I. Ander- −1 sen, J. Gorosabel, C. Bartolini, A. Guarnieri, et al., B 2 M 2 M˙ Science 283, 2069 (1999). L ≈ −13 L . 10 −1 1G M Myr A. H. Bridle, http://www.cv.nrao.edu/~abridle/ bgctalk (1998). (16) Z. Paragi, I. Fejes, R. C. Vermeulen, R. T. Schilizzi, R. E. Spencer, and A. M. Stirling, Astrophysics and Space Science Supplement 276, 131 (2001). 3. Conclusion M. J. Rees, in Black Holes and Relativistic Stars (1998), pp. 79–101. Astrophysical jets exist due to a precise combina- E. Buckingham, Phys. Rev. 4, 345 (1914). tion of electromagnetic, mechanic and gravitational L. Sedov, Similarity and Dimensional Methods in Me- processes, independently of the nature and mass of chanics (CRC Press, 1993), 10th ed. their central objects. D. Meier, in New Views on Microquasars (2002), pp. Here we report the dimensional relation between a 157–165. few important parameters that enter into the descrip- V. Trimble and M. J. Aschwanden, Publications of the tion of the formation of an astrophysical jet. ASP 116, 187 (2004). Of all our results, it is striking the fact that the jet R. Blandford, in Active Galactic Nuclei,editedbyT.- power is inversely proportional to the accretion rate L. Courvoisier and M. Mayor (Springer-Verlag, Les associated with it. This is probably due to the follow- Diablerets, 1990), Saas-Fee Advanced Course 20, ing. For a fixed value of the mass of the central object pp.
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