Editing Palermo scale (section)

== Scale ==

The Palermo scale was devised for astronomers to compare impact hazards at a technical level, rather than for the general public.<ref>{{cite journal |first1=Steven R. |last1=Chesley |first2=Paul W. |last2=Chodas |first3=Andrea |last3=Milani |first4=Giovanni B. |last4=Valsecchi |first5=Donald K. |last5=Yeomans |title=Quantifying the risk posed by potential Earth impacts |journal=[[Icarus (journal)|Icarus]] |volume=159 |issue=2 |pages=423––432 |date=October 2002 |doi=10.1006/icar.2002.6910 |bibcode=2002Icar..159..423C }}</ref> It was adopted at the meeting of the Working Group on Near-Earth Objects of the Scientific and Technical Subcommittee of the [[United Nations Committee on the Peaceful Uses of Outer Space]] which was held in [[Palermo]], Italy, on June 11–16, 2001.<ref>{{cite report |first1=David |last1=Morrison |first2=Andrea |last2=Milani |first3=Richard |last3=Binzel |display-authors=etal |title=Working group on near Earth objects |year=2003 |work=Transactions of the International Astronomical Union |volume=XXV A |pages=139–140 |publisher=IAU |editor-first=Hans |editor-last=Rickman |doi=10.1017/S0251107X00001358 }}</ref>

The scale compares the likelihood of the detected potential impact with the average risk posed by objects of the same size or larger over the years until the date of the potential impact. This average risk from random impacts is known as the background risk. The Palermo scale value, <math>P</math>, is defined by the equation:
:<math>P \equiv \log_{10} \frac {p_i} {f_B T}</math>
where
:*<math>p_i</math> is the impact probability
:*<math>T</math> is the time interval until the potential impact that is considered
:*<math>f_B</math> is the background impact frequency

The background impact frequency is defined for this purpose as:
:<math>f_B = 0.03\, E^{-\frac45} \text{ yr}^{-1}\;</math>
where the energy threshold <math>E</math> is measured in [[TNT equivalent|megatons]], and yr is the unit of <math>T</math> divided by one year.

For instance, this formula implies that the [[expected value]] of the time from now until the next impact greater than 1 megaton is 33 years, and that when it occurs, there is a 50% chance that it will be above 2.4 megatonnes. This formula is only valid over a certain range of <math>E</math>.

However, another paper<ref>{{cite journal |title=The flux of small near-Earth objects colliding with the Earth |author=P. Brown |display-authors=etal |journal=[[Nature (journal)|Nature]] |volume=420 |issue=6913 |pages=294–296 |date=November 2002 |bibcode=2002Natur.420..294B |doi=10.1038/nature01238 |pmid=12447433 |s2cid=4380864}}</ref> published in 2002 – the same year as the paper on which the Palermo scale is based – found a power law with different constants:

:<math>f_B = 0.00737 E^{-0.9} \;</math>

This formula gives considerably lower rates for a given <math>E</math>. For instance, it gives the rate for [[bolide]]s of 10 megatonnes or more (like the [[Tunguska explosion]]) as 1 per thousand years, rather than 1 per 210 years (or a 38% probability that it happens at least once in a century) as in the Palermo formula. However, the authors give a rather large uncertainty (once in 400 to 1800 years for 10 megatonnes), due in part to uncertainties in determining the energies of the atmospheric impacts that they used in their determination.

{| class="wikitable" style="text-align:center; font-size: 0.9em;"
|+Palermo Background Risk Chart
! rowspan="3" | Energy (MT)
! colspan="5" | Probability 
|-
! rowspan="2" | Once in<br/>this many years
! colspan="4" | At least once in one... 
|-
! decade
! century
! millennium
! million years
|-
| 0.1 || 5.28
| 87.73%
| >99.99%
| >99.99%
| >99.99%
|-
| 1 || 33.3
| 26.26%
| 95.24%
| >99.99%
| >99.99%
|-
| 10 || 210
| 4.65%
| 37.91%
| 99.15%
| >99.99%
|-
| 100  || 1,327
| 0.75%
| 7.26%
| 52.94%
| >99.99%
|-
| 1,000 || 8,373
| 0.12%
| 1.19%
| 11.26%
| >99.99%
|-
| 10,000 || 52,830
| 0.019%
| 0.19%
| 1.88%
| >99.99%
|-
| 100,000 || 333,333
| 0.003%
| 0.03%
| 0.3%
| 95.02%
|-
| 1,000,000 || 2,103,191
| 0.00048%
| 0.0048%
| 0.048%
| 37.84%
|-
| 10,000,000 || 13,270,239
| 0.000075%
| 0.00075%
| 0.0075%
| 7.26%
|-
| 100,000,000 || 83,729,548
| 0.000012%
| 0.00012%
| 0.0012%
| 1.19%
|-
| 1,000,000,000 || 528,297,731
| 0.0000019%
| 0.000019%
| 0.00019%
| 0.19%
|-
| <math>E</math> || <math>\frac{1}{f_B}=\frac{100}{3}E^{0.8}</math>
| <math>1-\left(1-f_B\right)^{10}</math>
| <math>1-\left(1-f_B\right)^{100}</math>
| <math>1-\left(1-f_B\right)^{1,000}</math>
| <math>1-\left(1-f_B\right)^{1,000,000}</math>
|}

For asteroids with multiple (<math>n</math>) potential impacts, the cumulative Palermo scale rating, <math>P_{cum}</math>, is the rating that can be calculated with the sum of the probability ratios of the individual potential impacts (each calculated with a <math>p_i</math> probability and a <math>T_i</math> time until potential impact), which can also be expressed as the logarithm of the sum of 10 raised to the <math>P_i</math> Palermo scale rating of the individual potential impacts:<ref name="palermo"/>

:<math>P_{cum} = \log_{10} \sum_{i=1}^{n} \frac {{p_i}} {f_B T_i} = log_{10}{\left(\sum_{i=1}^{n} 10^{P_i}\right)}</math>