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I know what 2.5 stand for in $2.5 \log_{10}$, but what does the number 5 in $5\log_{10}$ stand for and what application does it have.

Thus, why is the number a five and not a four, or a three, or something else?

2.5 stands for the logarithmic step in apparent magnitude. There must be something similar for the 5 in 5log10. Does it stand for the logarithmic step in absolute magnitude?

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    $\begingroup$ Check the definition of magnitudes and the distance modulus. 5 stands for the number 5. $\endgroup$ Nov 9 '21 at 15:08
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    $\begingroup$ I’m voting to close this question because it belongs on Mathematics SE. $\endgroup$ Nov 9 '21 at 15:12
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    $\begingroup$ You need to provide details about where these calculations are used. Usually, an equation is something = 5Log10(of something else). What are "something" and "something else"? As presented, 2.5 and 5 are multipliers. That is all we can say without knowing the full context. $\endgroup$
    – JohnHoltz
    Nov 9 '21 at 15:34
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    $\begingroup$ Assuming it refers to the linked posting: It's an empirical formula and 5 stands for the number 5 which happens to match whatever relation is explained in the study it comes from. $\endgroup$ Nov 9 '21 at 19:20
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    $\begingroup$ The relation between flux and magnitude is defined (5 magnitudes is a factor of 100 in brightness) and does not follow from anything physical. $\endgroup$ Nov 10 '21 at 8:06
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A magnitude is defined as $-2.5\log_{10}$ of a flux. But flux scales as the inverse square of a distance.

$$-2.5\log_{10}\left(\frac{k}{d^2}\right) = 5\log_{10}(d) -2.5\log_{10}(k)$$

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This appears to be a question directed at an answer to a previous question by the OP. https://astronomy.stackexchange.com/a/47296/26216

The 5log_10(...) appears to just be a multiplier the author of the paper cited in the aforementioned answer found to describe the visible magnitude of Neptune. Sorry, but that's probably the best answer you'll get... There may be something special about its formulation, or it could just be a curve that best fitted the data.

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