Using the force constant in equations

[nightcleaner]

Hi Marcus

Just thought I'd point out that Cambridge Handbook of Physics Formulas, p 183, uses solar mass in the BH evaporation time formula. I'll try to transcribe in into LaTex here.

\tau_e = \frac{M^3}{(M_@)^3}x10^66 years

Well that's pretty good, except that it is 10^66 , and here the ampersand is to represent a circle with a dot in it (which means, in CHOP, one solar mass), and the equivalence sign here is just a wavy line in CHOP, to indicate "approximately equal".

"one solar mass" is also used in the CHOP formulae for Schwartzchild radius, Chandrasekhar limit, and black hole temperature.

I guess they follow this convention because a solar mass is a convenient unit when talking about black holes as cosmological objects.

Solar mass in Planck units? One solar mass in Wiki is 1.9891 x 10^30 kg

I have Planck mass as .434 x 10^-8 kg

I get 4.58x10^38 Planck units for solar mass.


nc

 

[marcus]

Wonderful! you hit Cambridge paydirt, or at least an isolated nugget. I will look at it and compare.

Hi Marcus

Just thought I'd point out that Cambridge Handbook of Physics Formulas, p 183, uses solar mass in the BH evaporation time formula. I'll try to transcribe in into LaTex here.

\tau_e \sim \frac{M^3}{(M_\odot)^3} \times 10^{66} \text{ years}

Well that's pretty good, except that it is 10^66 , and here the ampersand is to represent a circle with a dot in it (which means, in CHOP, one solar mass), and the equivalence sign here is just a wavy line in CHOP, to indicate "approximately equal".

"one solar mass" is also used in the CHOP formulae for Schwartzchild radius, Chandrasekhar limit, and black hole temperature.

I guess they follow this convention because a solar mass is a convenient unit when talking about black holes as cosmological objects.

Solar mass in Planck units? One solar mass in Wiki is 1.9891 x 10^30 kg

nc

EDIT: put in \sim or \approx
for one squiggle or two squiggles

 

[marcus]

you might want to look at how I edited your LaTex formula


\tau_e \sim \frac{M^3}{(M_\odot)^3} \times 10^{66} \text{ years}

\tau_e \approx \frac{M^3}{(M_\odot)^3} \times 10^{66} \text{ years}

about their formula, it is just approximate, but looks very handy and useful.

we know that the evaporation time varies as the cube of the mass (that is how our formula went too)

so you can pick a convenient mass like the solar, and calculate the evap. time just for THAT, using some more basic messy derivation.

then suppose it comes out approximately E66 years (which it probably does)

from that point onwards you can use their simple formula which uses the solar mass as a point of reference.

if some hole is TWICE the solar mass, well it goes by cubes so the evaporation time would be EIGHT TIMES AS LONG and we know that the evap time for the solar mass is (approximately) E66 years, so it would be 8E66 years for a two solar mass hole

to verify the Cambridge formula, all I really need to do is check that it is (approx.) correct for one solar mass. we could do that, maybe i will

 

[marcus]

I get 4.58x10^38 Planck units for solar mass.

Yes! let's cube that and multiply by 80/pi

that will do it

when I cube 4.58 I get 96
so cubing 4.58E38 gives 96E114

multiplying 96 by 80/pi and E114 gives 2.44 E117

This is 2.1 X 10⁶⁷ years

But they say the evaporation time for a solar mass hole is a flat 10⁶⁶ years. I get a factor of 21 larger.

Well! their formula is off some!
We should write to them and point this out, maybe they can correct it in the next edition (or maybe they do not care and only want something that is correct to an an order of magnitude for something like this)

Also I will double check my formula to make sure IT is not off by a factor of two.

this is embarrassing. I can't find any mistake in my formula or arithmetic. Also I checked the Wiki article and they agree more or less with me, rather than Cambridge. Wiki gives some formulas in "common units" permitting one to estimate evaporation times. and they give an approximate time for the sun which is E67 years (rather than what Cambridge says E66).

 

[nightcleaner]

Hi Marcus

Yes I saw how you corrected my LaTex, thanks. Odot and curley brackets. Who would have thought. But I see you didn't put in the little squiggle for "Approximately equal."

nc

 

[marcus]

But I see you didn't put in the little squiggle for "Approximately equal."

I forgot that, earlier, but just saw your post and went back and put in a squiggle or a wavy equals sign.

I will try to get a link to some pages on LaTex that has these extra symbols. It is at some page of the PF thread on LaTex which however is now so long that it is laborious to look through it.

Oh good! The very first post of that thread has the links we need:
https://www.physicsforums.com/showthread.php?t=8997

here are some links given in that thread, I quote from Warren's post.

"A pdf file of the most useful LaTeX commands, symbols, and constructs is provided here:"

https://www.physicsforums.com/misc/howtolatex.pdf

"More symbol reference:"

http://amath.colorado.edu/documentation/LaTeX/Symbols.pdf

 

[nightcleaner]

Thanks Marcus this is great. I'll add these to my favorites list.

R.

 

[marcus]

Nightcleaner seems still to be at his cabin and we hear from him only sporadically. Maybe it is a question of batteries. Or on the other hand maybe he has been overwhelmed by the silence and cannot think of anything to say.

I have gotten quite comfortable with using units based on the force constant. A recent paper by Padmanabhan called them "rationalized Planck units", which may be a good designation. At any rate they now seem quite natural.

Out in the garden the sun is so bright I've taken to wearing a broad-brimmed straw hat. It is cold in the house and tempting to go out and let the sun warm my bones.

The raven nesting in the palm tree, or on lookout atop an even taller redwood, always calls three times
the pitch (which i determined by rushing indoors to compare it with keys on the piano) is the D beside middle C.
Accordingly the raven's frequency is (1/2)E-40 natural
or it might be D#, which would be just 6 percent higher----0.53E-40.

In natural terms the sunlight power per unit area is 6E-117. I finally got used to that. Think of it whenever, outdoors on a clear day, I see and feel the light.

have come to appreciate pressure 14E-107 which I can't feel but know is there. Know it's there because the fishpond would otherwise be simultaneously freezing and boiling, and we would have freeze-dried goldfish. That pressure of E-107 is nearly "pound per inch", so I know to what level, in "rationalized Planck" terms, I just inflated the tires.

Everything seems in order here on the immediate grounds, though zealots destroy the nation's moral standing and economy, still the sky is beautiful and all is well.

Not having heard from Richard the NC, I will suppose him still out in the woods, alone in his cabin, eating freeze-dried goldfish, since they are much easier to carry when you have to pack your food in.