The Apollo asteroids'

*H mag* is very much related to their inclination

*i* and perihelium (

* a*(1-e) *).

As the

*Tisserand's *parameter is related to the orbital parameters (

*a,e,i*), one may wonder whether the

* H mag* is also related to the

*Tisserand* distribution.

At first, I had been trying to look at the relation between

*H mag* and

*Tisserand's *parameter with respect to Jupiter.

The relation seems to be very weak, so I have tried to see if a different choice for the semi-major axis of the perturbing body can provide a stronger relationship.

The conclusion is that the stronger relation is found with a perturbing body semi-major axis equal to 1.01 AU (more or less ...Earth).

The analysis has been done with the

package R (see R Core Team (2016). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ).

In the following sections, taking into account the Apollo asteroids, I will describe:

- data preparation
- H mag distribution
- H mag versus inclination
- H mag versus perihelium
- H mag versus Tisserand's parameter with respect to Jupiter
- investigation about a generic Tisserand's parameter
- the special case of Tisserand's parameter for a perturbing body with semi-major axis = 1.01 AU

**Data preparation **

After that, I read the file apollo.csv as a dataframe in R with the below instruction:

*p<-read.csv("apollo.csv")*

With the head and tail instructions you can look at the first rows and at the last rows of your data frame:

*> head(p)*

a e i om w H t_jup

1 1.078078 0.8268746 22.826863 88.012367 31.3817 16.90 5.298

2 1.245393 0.3353925 13.337454 337.216252 276.8695 15.60 5.075

3 1.367258 0.4359054 9.380703 274.339634 127.0826 14.23 4.716

4 1.470241 0.5599397 6.352955 35.739118 285.8483 16.25 4.415

5 2.259355 0.6061736 18.399653 346.487607 267.9952 15.54 3.298

6 1.460856 0.6146128 22.212986 6.640199 325.6343 14.85 4.336

*> tail(p)*

a e i om w H t_jup

7855 1.313828 0.3839697 10.811349 277.5693 123.8834 22.621 4.872

7856 1.355240 0.5707721 6.501750 294.3308 251.8823 24.329 4.672

7857 2.449819 0.6928541 1.086150 133.4694 231.7488 25.417 3.113

7858 1.306445 0.2387369 14.003415 288.4028 329.4405 23.046 4.927

7859 2.693641 0.6526003 10.463092 133.2149 213.9876 24.597 3.004

7860 2.804018 0.6672408 4.724818 183.5757 234.1325 20.400 2.946

So we read a total of 7860 rows.

With the summary instruction you can have a look at the data:

*>summary(p)*

a e i om

Min. : 1.000 Min. :0.02097 Min. : 0.02262 Min. : 0.019

1st Qu.: 1.294 1st Qu.:0.36393 1st Qu.: 4.24058 1st Qu.: 80.208

Median : 1.643 Median :0.51276 Median : 8.44792 Median :169.915

Mean : 1.724 Mean :0.49672 Mean : 12.52270 Mean :170.999

3rd Qu.: 2.090 3rd Qu.:0.62793 3rd Qu.: 17.46998 3rd Qu.:251.462

Max. :17.827 Max. :0.96910 Max. :154.37505 Max. :359.937

w H t_jup

Min. : 0.0348 Min. :12.40 Min. :1.316

1st Qu.: 92.4286 1st Qu.:20.00 1st Qu.:3.444

Median :189.4941 Median :22.60 Median :4.088

Mean :182.7822 Mean :22.57 Mean :4.186

3rd Qu.:272.6443 3rd Qu.:25.17 3rd Qu.:4.878

Max. :359.9882 Max. :33.20 Max. :6.064

**H mag distribution**

With the following instructions you can display a rough density plot to have an idea about the data:

*> library(ggplot2)*

> ggplot(p,aes(H))+geom_density()

**H mag versus inclination**

The curious density shape shown above has an easy explanation.

Just take a look at the inclination quartiles (in short, I call them i_quartiles):

*> i_quartiles<-cut(p$i,fivenum(p$i),include.lowest=T)*

*> ggplot(p,aes(H,col=i_quartiles))+geom_density()*

*> summary(i_quartiles)*

[0.0226,4.24] (4.24,8.45] (8.45,17.5] (17.5,154]

*1965 1965 1965 1965*

So it seems that inclination alone is a good explanation for the H mag distribution shown above. Apollo asteroids belonging to the 3rd and 4th inclination quartiles have, by definition, a greater inclination: it turns out that they are brighter than those belonging to the 1st and 2nd inclination quartiles.

**H mag versus perihelium**
In this case, we calculate

* q = a*(1-e)* and after that we find the perihelium quartiles:

*> p$q<-p$a*(1-p$e) *

> q_quartiles<-cut(p$q,fivenum(p$q),include.lowest=T)

> summary(q_quartiles)

[0.0707,0.694] (0.694,0.858] (0.858,0.954] (0.954,1.02]

1965 1965 1965 1965

*> ggplot(p,aes(H,col=q_quartiles))+geom_density()*
Apollo asteroids belonging to the 3rd and 4th perihelium quartiles
have, by definition, a greater perihelium: it turns out that they are darker than those belonging to the 1st and 2nd perihelium quartiles.

**H mag versus Tisserand's parameter with respect to Jupiter**

We can run the following instructions to get a rough scatter plot.

*> ggplot(p,aes(H,t_jup))+geom_point()*

This scatter plot is not particularly informative, it is hard to see any meaningful patterns!

So let's use the same approach seen before: let's make a plot based on T-Jupiter quartiles:

*> tjup_quartiles<-cut(p$t_jup,fivenum(p$t_jup),include.lowest=T)*

> summary(tjup_quartiles)

[1.32,3.44] (3.44,4.09] (4.09,4.88] (4.88,6.06]
* 1967 1965 1964 1964 *
Similarly, we can draw a boxplot. Just run these commands:

*ggplot(p,aes(tjup_quartiles,H,fill=tjup_quartiles))+\*
*geom_boxplot()+\** *
*scale_x_discrete(labels=c("Q1","Q2","Q3","Q4"))*

The Apollo asteroids belonging to the 1st T-jupiter quartile are brighter than the others.

A part from this, the distributions of the Apollo asteroids belonging to the 2nd, 3rd and a little less the 4th T-Jupiter quartiles are not so different as far as the median value is concerned.

We can find a rough numerical measure for the (not so great) difference between the various H mag distributions: we use R to calculate the spearman correlation.

*> cor(p$H,p$t_jup,method="spearman")*
*[1] 0.2268667*
I think the Spearman correlation in this case is better than the traditional Pearson correlation because we are not really hoping to find a linear relation, we would be happy to find a monotonic one.

**Investigation about a generic Tisserand's parameter**
Given the above result, one can wonder: what happens if we used a different Tisserand's parameter? Would we get a stronger correlation?

In order to answer, we can define two user defined functions:

__Function1__
*func_t<-function(ap,a,e,i){ap/a+2*(a/ap*(1-e^2))^(1/2)*cos(i/180*pi)}*
This first function just implements the

Tisserand formula:

__Function2__
*func_t_h_spea<-function(ap)\{p$ts<-func_t(ap,p$a,p$e,p$i);return(cor(p$H,p$ts,method="spearman"))}*
This second function calls the first one and after that, once the Tisserand values are calculated for every asteroid, it returns the correlation coefficient between H and the generic Tisserand's parameter (given

*ap* as input).

Now we can vary the semi-major axis

*ap* of the perturbing body in a range of our choice ...let's say

*[0<ap<30] AU* and plot the graph showing the variation of the correlation factor.

*>therange<-seq(0.01,30,0.01)*

>plot(therange,sapply(therange,function(x) {func_t_h_spea(x)}),type='l',xlab="ap used for tisserand param",ylab="Spearman correlation H vs Tisserand's param")
With the order instruction we can search for the exact

*ap* value that maximizes the correlation;

*> tail(order(sapply(therange,function(x) {func_t_h_spea(x)})),1)*

[1] 101

> therange[101]

[1] 1.01
We get

**ap = 1.01 AU**
**The special case of Tisserand's param with ap = 1.01 AU**
Given the above result, we can calculate the quartiles of the Tisserand's parameter with respect to a perturbing body with semi-major axis 1.01 AU:

*> ap=1.01*

>p$ts<-func_t(ap,p$a,p$e,p$i);p$tsquartile<-factor(cut(p$ts,fivenum(p$ts), include.lowest = T))
*>summary(p$tsquartile)*

[-1.29,2.59] (2.59,2.79] (2.79,2.9] (2.9,3]
* 1965 1965 1965 1965*
Then, we can see the result in the usual graphical form:

*> ggplot(p,aes(H,col=tsquartile))+geom_density()*
It seems that our search for a better/stronger correlation has been successful: now we can observe a sharper separation between the various H distributions.
The same result is more clear with a boxplot:
*ggplot(p,aes(tsquartile,H,fill=tsquartile))+\*
*geom_boxplot()+\** *
*scale_x_discrete(labels=c("Q1","Q2","Q3","Q4"))+ggtitle("H versus Tisserand's param for ap=1.01")*
In conclusion, there are a lot of outliers in the extreme quartiles but the middle half of the H mag distributions are now much better separated than when we used Tisserand's parameter with respect to Jupiter.
Cheers,
Alessandro Odasso