## Friday, October 14, 2016

### Extreme TNOs - H mag

Let's apply the same approach described in the previous posts to the TNOs.

I downloaded some days ago a list of 2298 TNOs from the Horizons Web-Interface.

After reading it in an R dataframe, I have plotted the Spearman correlation versus Tisserand parameter with respect to a generic body with ap varying between 20 AU and 1000 AU. Important: I added a filter so to include ONLY asteroids with T <=3 ... this implies that the numerosity of the asteroids is not constant along this plot.

After some trials, it seems to me that there is a moderate correlation (better: anti-correlation) for ap = 324 AU (red vertical line).
This is visible in a rough but clear way if you just make a boxplot showing the different H mag distributions in the first and second half of the Tisserand range.

It must be said that the total number of TNOs in this plot is only 45: so I am not sure whether this is significant.

It must also be said that these 45 asteroids are very interesting: among them we can find Sedna and other famous Extreme TNOs that are being studied extensively because their orbits might have been shaped by an unknown massive planet waiting to be discovered.

We can note that the 45 asteroids have a>136 aU and e>0.677
> tapply(z$a,list(z$T324),min)
FALSE      TRUE
30.12661 136.09255
> tapply(z$e,list(z$T324),min)
FALSE      TRUE
0.0000000 0.6776645

Another interesting aspect is described below.

45 TNOs - H Mag depends on Tisserand vs outer planets
Let's calculate the Tisserand parameter Tn with respect to Neptune.
After that we can flag every asteroid TNOs as Tn_le_3 or Tn_gt_3 depending on whether Tn is less than 3 or greater.

Furthermore: we repeat this process for Uranus, Saturn and Jupiter.
In the end, every asteroid is flagged as follows:
• Neptune:  Tn_le_3 or Tn_gt_3
• Uranus - Tu_le_3 or Tu_gt_3
• Saturn -  Ts_le_3 or Ts_gt_3
• Jupiter - Tj_le_3 or Tj_gt_3
Of course, following the same convention, all our 45 TNOs have T324_le_3: this is how we chose them.

This is the boxplot showing H mag of the 45 TNOs depending on the combination of the above flags:
As you see, something quite peculiar (at least for me!) is going on:
• the brightest TNOs are those who have any Tisserand parameter with respect to outer planets greater than 3.
• when Neptune comes in (I mean: Tn<le_3), the asteroids get darker
• when Neptune and Uranus come in ... the asteroids get darker
• when Neptune, Uranus and Saturn come in ... same as above ... only 1 asteroid.
• when also Jupiter come in ... we find the darkest TNOs of this group of 45 TNOs.

This is something that I can not explain:
• is this an expected / known result?
• is it a ... statistical fluke?

Cheers,
Alessandro Odasso

## Tuesday, October 11, 2016

### Amor - a summary plot of H vs Tisserand's parameter - Tp < 3

This post continues from the previous one.

Let's take again the plot H  mag vs Tisserand's parameter with respect to Earth (ap=1 AU).

ONLY those Amors with TEarth<3

We have already seen this plot, we just add a few labels to count them:

We have also seen that most of these Amors have also Tisserand's parameter with respect to Jupiter, Venus and Mars <  3 (actually, I refer to it as "Mars" but the exact value for which there was a relative greater correlation between H mag and Tisserand's parameter was ap = 1.65 AU).

A possible question is:how are these asteroids distributed  in the various TEarth quartiles?

For graphical purposes, I introduce the following abbreviations:
• Tm = TRUE - Tisserand's parameter with respect to Mercury < 3
• Tv = TRUE - Tisserand's parameter with respect to Venus < 3
• Tma = TRUE - Tisserand's parameter with respect to Mars (better: ap =1.65 aU) < 3

This is the result:

I do not know enough about statistic to say that the above is really significant, but I can give you my feeling.
It seems to me that:
• in TEarth quartile Q2, Q3 and Q4: those Amors having Tma < 3 (red) tend to be brighter than the others not having Tm or Tv < 3 (green)
• in TEarth quartile Q1 and Q2: those Amors having Tma < 3 and Tv < 3 (yellow) tend to be slightly darker than those having only Tma < 3 (red)
• in TEarth quartile Q1: those 79 Amors having Tm<3 (and also Tv and Tma <3) tend to be the brightest.

Below, you can see a table showing the numerosity (as also shown in the labels displayed in the plot) and the median and sd of every group:

Numerosity
> t(tapply(pe$H,list(pe$tsquartile,pe$Tm:pe$Tv:pe$Tma),length)) [-0.15,2.82] (2.82,2.93] (2.93,2.97] (2.97,3] FALSE:FALSE:FALSE NA 4 33 79 FALSE:FALSE:TRUE 213 871 884 838 FALSE:TRUE:FALSE 1 2 NA NA FALSE:TRUE:TRUE 624 40 NA NA TRUE:FALSE:FALSE NA NA NA NA TRUE:FALSE:TRUE NA NA NA NA TRUE:TRUE:FALSE NA NA NA NA TRUE:TRUE:TRUE 79 NA NA NA Median > t(tapply(pe$H,list(pe$tsquartile,pe$Tm:pe$Tv:pe$Tma),median))
[-0.15,2.82] (2.82,2.93] (2.93,2.97] (2.97,3]
FALSE:FALSE:FALSE           NA       23.85        24.0     24.6
FALSE:FALSE:TRUE          19.7       20.60        22.9     23.6
FALSE:TRUE:FALSE          23.6       24.15          NA       NA
FALSE:TRUE:TRUE           20.1       23.10          NA       NA
TRUE:FALSE:FALSE            NA          NA          NA       NA
TRUE:FALSE:TRUE             NA          NA          NA       NA
TRUE:TRUE:FALSE             NA          NA          NA       NA
TRUE:TRUE:TRUE            19.2          NA          NA       NA

Standard deviation
> t(tapply(pe$H,list(pe$tsquartile,pe$Tm:pe$Tv:pe$Tma),sd)) [-0.15,2.82] (2.82,2.93] (2.93,2.97] (2.97,3] FALSE:FALSE:FALSE NA 0.9832238 1.984962 2.098781 FALSE:FALSE:TRUE 2.153851 2.4985271 2.592066 2.609018 FALSE:TRUE:FALSE NA 1.2020815 NA NA FALSE:TRUE:TRUE 2.234240 2.1410988 NA NA TRUE:FALSE:FALSE NA NA NA NA TRUE:FALSE:TRUE NA NA NA NA TRUE:TRUE:FALSE NA NA NA NA TRUE:TRUE:TRUE 1.967811 NA NA NA Cheers, Alessandro Odasso ## Saturday, October 8, 2016 ### Amor - H mag vs Tisserand (ONLY Tp < 3) I refer to my previous post. Following some comments from Alan Harris, I have re-drawn the relation H mag vs Tisserand with respect to a planet with generic semi-major axis ap taking into account ONLY those asteroids that have Tp < 3. Amor - Method 1 Method based on Spearman correlation. Having filtered only those asteroids that have Tp < 3, I have to restrict the range where the semimajor axis ap can vary: 0.35 AU <= ap <= 3 AU For the purpose of the next plot, this is a table showing the orbital parameters of the planets: Semimajor Axis (AU) Orbital Period (yr) Orbital Speed (km/s) Orbital Eccentricity (e) Inclination of Orbit to Ecliptic (°) Rotation Period (days) Inclination of Equator to Orbit (°) Mercury 0.3871 0.2408 47.9 0.206 7.00 58.65 0 Venus 0.7233 0.6152 35.0 0.007 3.39 -243.01* 177.3 Earth 1.000 1 29.8 0.017 0.00 0.997 23.4 Mars 1.5273 1.8809 24.1 0.093 1.85 1.026 25.2 Jupiter 5.2028 11.862 13.1 0.048 1.31 0.410 3.1 Saturn 9.5388 29.458 9.6 0.056 2.49 0.426 26.7 Uranus 19.1914 84.01 6.8 0.046 0.77 -0.746* 97.9 Neptune 30.0611 164.79 5.4 0.010 1.77 0.718 29.6 This is the plot showing the correlation between H mag and Tp calculated for various values of semimajor axis ap expressed in AU: I have added the following vertical lines, showing the ap of the planets: • blue line - Mercury • green line - Venus • red line - Earth • magenta line - Mars I wonder whether the planets are truly responsible for the peaks that we see in this plot. This is probably true for Earth, because the peak is located at ap = 1.0 AU Not clear if we can claim that this is true for Mercury and Venus. In the case of Mars, this is less convincing: the peak next to Mars is not located at ap=1.52 AU but slightly greater. Boxplot for ap=0.3871 AU we only have 79 asteroids, with tp < 3, belonging to these Tp quartiles. > summary(p1$tsquartile)
[-0.299,2.34]   (2.34,2.58]   (2.58,2.89]      (2.89,3]
20            20            19            20

Boxplot for ap=0.7233  AU
In this case we have 746 asteroids with T<3
> summary(p1$tsquartile) [-0.198,2.68] (2.68,2.85] (2.85,2.94] (2.94,3] 187 186 187 186 Boxplot for ap=1.0 AU In this case we have 3668 asteroids with T<3 > summary(p1$tsquartile)
[-0.15,2.82]  (2.82,2.93]  (2.93,2.97]     (2.97,3]
917          917          917          917

Boxplot for ap=1.65 AU
It turns out that the peak next to Mars semimajor axis is located at about ap= 1.65 AU
In this case we have 5617 asteroids with T<3
> summary(p1$tsquartile) [-0.0765,2.66] (2.66,2.75] (2.75,2.83] (2.83,3] 1405 1404 1404 1404 Finally, I would like to see if the method2 based on Chi-squared test gives a similar answer. Amor - Method2 Method based on Chi-squared statistic (see previous post): In this case the effect of Mercury (if real) can not be seen. Much more clear, if real, the effect of Earth and Mars ... some doubts about Venus. Cheers, Alessandro Odasso ## Thursday, October 6, 2016 ### Amor asteroids - H mag vs Tisserand parameter As seen in the previous blog where I focused on Apollos, I am now going to do the same for Amors: look for any relation between H mag and the Tisserand parameter with respect to a generic body with semi-major axis ap. The objective is to see what happens as ap changes in a given range. Two simple ways to do this: • method1: calculate the (spearman) correlation between H mag and Tp, plot the result for many values of ap. • method2: divide the H mag and Tp range into quartiles; then, calculate the Chi-squared statistics on the resulting contingency table. Plot the result for many values of ap. These two methods give almost the same result in the case of Apollo asteroids. What is very strange, at least for me, is that these two methods give a different answer in the case of Amor asteroids. In the following sections, I will describe in more detail: • Apollo - method1 • Apollo - method2 • Amor - method1 • Amor - method2 • appendix (more graphs showing H mag vs others orbital parameters) Apollo - method1 We calculate the spearman correlation and plot it against many values of ap. We search for minimum or maximum. This method and the graphical result has been described in the previous blog. Apollo - method2 I did not applied this approach in the previous blog, so I describe it in detail and I show the result. First, we divide the H mag into quartiles (H-quartiles). Then we calculate the Tp param for all asteroids and we divide the resulting range into quartiles (Tp-quartiles). After that, we build a contincency table showing H-quartiles vs Tp-quartiles and, finally, we calculate the Chi-squared statistic. The idea is just this: let's imagine for a moment that there is absolutely no relation between H mag and Tp. In such a case the Chi-squared statistic would be zero because there would be no difference between the observed values and the expected values, the latter being, by definition, the values that would be displayed when no relation exists. On the contrary, the greater the Chi-squared statistic and the greater the difference of H mag distributions between the Tp quartiles. In order to implement all the above in R, besides reading the orbital parameters values downloaded from Horizons Web-Interface in a dataframe, we need a couple of user defined functions: func_t function(ap,a,e,i){ap/a+2*(a/ap*(1-e^2))^(1/2)*cos(i/180*pi)} The above function just implements the Tisserand formula. func_t_h_chisq function(ap){p$ts<-func_t(ap,p$a,p$e,p$i);p$tsquartile<-factor(cut(p$ts, fivenum(p$ts), include.lowest = T));return(chisq.test(with(p,table(tsquartile,hquartile)))\$statistic)}

All asteroids parameters are in a dataframe called p.
The above function calculates the Tp values (calling the first function) for all asteroids in the dataframe p: then it calculates the quartiles (cut and fivenum instructions), the contincency table (table instruction) and the Chi-squared statistic.

In order to see the result, all we have to do is define a range for ap (lets say 0 <= ap<=30 AU), loop calling the function and plot it:

therange<-seq(0.01,30,0.01)
plot(therange,sapply(therange,function(x) {func_t_h_chisq(x)}),type='l',xlab="ap (tisserand param)",ylab="H-quartiles vs Tp-quartiles - Chi-squared statistic",main="Apollo asteroids")

The maximum correlation is found for ap=1.03 AU
I think that this result is almost the same we achieved with method1 where we got ap = 1.01 AU.
More important: the plot seems smooth, there is a clear maximum.

Amor - method1
See previous blog for R instructions.
This is the plot showing how the correlation varies depending on the choiche of ap:
The maximum correlation in the case of Amors is found for ap = 2.07 AU
However, the overall behaviour is "flat" compared to the bigger variations that we saw in the case of Apollos.
In fact, the boxplot shows some differences (but not so many!) in the distribution of H depending on Tp quartiles:

Amor - method2
Following the same approach already described for Apollo asteroids, in order to see the result, all we have to do is define a range for ap (lets say 0 <= ap<=30 AU), loop calling the function and plot it:

therange<-seq(0.01,30,0.01)

plot(therange,sapply(therange,function(x) {func_t_h_chisq(x)}),type='l',xlab="ap (tisserand param)",ylab="H-quartiles vs Tp-quartiles - Chi-squared statistic",main="Amor asteroids")

This is surprising!
There is a relative minimum for ap = 1.29 AU
There are two relative maxima for ap = 1.0 AU and ap = 1.94 AU

We do not get once again the value ap=2.07 AU that was found before using the correlation method.

Let's see the corresponding boxplots to see if we have smaller differences for ap = 1.29 AU and much greater differences for ap = 1.0 AU and ap = 1.94 AU

boxplot ap=1.29 AU (minimum)
This boxplot below is done for ap=1.29 AU and, in fact, the differences of H distributions depending on different quartiles are small (with a notable exception for the 1st Tp quartile, where Amor asteroids are brighter.

boxplot ap=1.00 AU (maximum)
This boxplot below shows what happens for ap =1 .00 AU where we can see more differences compared to the previous one (although it seems that Q1 and Q4 are more similar):

boxplot ap=1.94 AU (maximum)
This boxplot below shows what happens for ap =1 .94 AU

In this other case, there is still some difference although this time Q2 and Q3 are more similar.
The trend seems consistent with what we have see before for ap=2.07 that was found using the correlation method.

appendix
As already done for Apollo in the previous blog, let's see other graphs for Amors:
• H mag vs inclination
• H mag vs perihelium

H mag vs inclination
Higher inclined Amors are brighter

H mag vs perihelium
Amors with greater perihelium (Q4 quartile) are brighter.
This is the opposite compared with Apollo belonging to their Q4 quartile that, on the contrary, were typically darker.

Cheers,
Alessandro Odasso