As I write this, I am sitting in Sheffield’s fine Victorian railway station, on my way home from visiting the astronomers here. Yesterday I sang for my supper, giving a double bill seminar on the radio background and on the big blue bump. Supper duly followed, and was exceeding pleasant. The Sheffield group is a small but lively one. As normal, the postdocs were worried about job prospects and the academics were moaning about writing exam questions. To quote P.C. : “I love my job but I hate writing exam questions.” With you there, Professor C.
The Crowther mentioned my recent blog post about how the Universe is almost empty. He said he likes to set his classes ballpark estimate exercises. He tried one out on me. You can have a go too. Its quite good to start by using your instinct to make a guess, before gathering a few facts to do the quick mental calculation. That way you can get your frisson of surprise. Suppose, said Paul, you take the material of the Earth and stretch it out from here to the Sun – how wide would that rope be ? His students guess a wide range of answers, but usually around a mm. Give it a go.
Here is another one, for which I will give you the answer – how powerful is a one kg accreting black hole ? As we all know, quasars are immensely luminous because they have supermassive black holes at their hearts – those accreting black holes are the most efficient energy sources we know. We are talking big numbers. At the Eddington limit, that billion solar mass black hole can be radiating up to 10^47 erg/s. When I first came to giving an undergraduate course with some of this stuff in, I felt duty bound to do things in SI units. So I got a formula for the Eddington limit : L_Edd=6.37M. Wuh ? A one kg black hole gives me 6 Watts ? My electric fire can do better than that ! My electric fire is only a few kg, but it gives me 2 kW.
Then comes the epiphany. You realise that accretion is not the slightest bit efficient per unit of mass of accretor ; what is impressive is the energy per unit mass of the accretee. To get 2kW out and so heat your bedsit, you need a black hole of at least 314 kg – about as big as two motorbikes. However, once you have it, the rate you have to burn fuel as it were, i.e. the rate at which you need to drop matter into the black hole, is about 2 x 10^-13 kg/s; one kg of fuel would last you roughly 140,000 years… I now leave it as an exercise for the reader how long a kg of coal would give that same 2kW. Scaling up to the most luminous quasars, you need to eat about 20 solar masses a year. Peanuts.
Then it made me think again about that electric bar fire. Its not really producing 2kW all by itself. It is plugged in and sucking that energy out of a giant power station miles away. Its really a similar story. First you need a vast power factory, and all the surrounding infrastructure. Only once you have it can you burn some fuel and give the illusion of that tiny fire producing energy. Likewise, before you can get that accretion energy goodness, you somehow have to assemble those billion solar masses. But thats another story…