---
title: "Losmandy GM811G Unguided Tracking"
author: "Michael A. Covington"
date: "2023 December 4"
date-format: "YYYY MMMM DD"
format: pdf
editor: visual
---

# Introduction

This is a test of the unguided tracking performance of my GM811G in early November, 2023, aimed high in the sky, using an iOptron iGuider (120mm focal length, 6.45 arc-seconds per pixel) with PHD2 (guiding corrections turned off).

# Data

I extracted the relevant portion from the PHD2 guide log.  It is a CSV file and looks like this:

```
Frame,Time,mount,dx,dy,RARawDistance,DECRawDistance,RAGuideDistance,DECGuideDistance,RADuration,RADirection,DECDuration,DECDirection,XStep,YStep,StarMass,SNR,ErrorCode
1,1.749,"Mount",-0.007,0.075,-0.074,-0.016,0.000,0.000,0,,0,,,,1527,27.11,0
2,3.331,"Mount",0.490,-0.037,-0.008,0.491,0.000,0.491,0,,0,,,,1548,27.44,0
3,4.914,"Mount",0.395,-0.125,0.088,0.407,0.000,0.407,0,,0,,,,1497,26.96,0
4,6.504,"Mount",0.502,0.031,-0.077,0.495,0.000,0.495,0,,0,,,,1498,26.91,0
5,8.095,"Mount",0.141,-0.116,0.102,0.154,0.000,0.154,0,,0,,,,1473,26.62,0
6,9.647,"Mount",0.192,-0.101,0.082,0.202,0.000,0.202,0,,0,,,,1499,26.93,0
```
with, of course, many lines of data following.

# First look at the data

```{r}
d = read.csv("GuideLogExtract.csv")
dd = data.frame(Frame=d$Frame, RA_Error=6.45*d$RARawDistance)
# 6.45 arcsec per pixel is in calibration data from PHD2
```

```{r}
require(ggplot2)
ggplot(dd, aes(Frame,RA_Error)) +
         geom_line()
```
Clearly, something bumped the telescope somewhere around the 20th frame, so let's cut off the first 25 frames and try again:

```{r}
ddd = dd[seq(25,nrow(dd)),]
ggplot(ddd, aes(Frame,RA_Error)) +
         geom_line()
```

Now we see a steady downward trend, caused by polar axis misalignment,
flexure, refraction, or some combination of the three,
and we need to remove it.

# Detrending

We can fit a line to the data using linear regression:

```{r}
model = lm(RA_Error ~ Frame, data=ddd)
ddd$Trend = predict(model, newdata=ddd)
```

Let's look at it:

```{r}
ggplot(ddd, aes(Frame,RA_Error)) +
         geom_line() +
         geom_line(color="red",aes(Frame,Trend))
```

That looks good.  Now subtract it:

```{r}
ddd$Detrended_RA_Error = ddd$RA_Error - ddd$Trend
ggplot(ddd, aes(Frame,Detrended_RA_Error)) +
  geom_line() +
  ylim(c(10,-10))
```

That is the quantity we actually want to study.  It is about 10
arc-seconds peak-to-peak.  If it were a sine wave, its RMS value
would be $10 / (2 \sqrt{2}) = 3.5$ arc-seconds.  But in fact most of
it is flatter than a sine wave, so let's do the calculation that we
set out to.

The RMS error is
simply the standard deviation:

```{r}
sd(ddd$Detrended_RA_Error)
```

Right, a whopping 1.62 arc-seconds RMS!  That is definitely good enough for
my 200-mm lens (4-arc-second pixel size) and probably also for my AT65EDQ
(about 2-arc-second pixel size).  Drift due to misalignment and flexure will
affect the image before tracking roughness does.  And it's supposed to get
even better when I upgrade to Gemini Level 6.