Interesting things about Voyager 1 and 2 launches

There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.

--Mark Twain

It's amazing how much we can do with such a small amount of data. The following analysis is based solely on the dates of launch and arrival of the Voyager spacecrafts, and an ephemeris of the planets.

Voyager 2 was launched on 1977-08-20T14:29:00Z, while Voyager 1 was launched later on 1977-09-05T12:56:00Z. Voyager 1 passed Voyager 2 on the way out (the exact time depends on how you define "pass") and arrived at Jupiter on 1979-03-05T12:05:26TDB, while Voyager 2 arrived on 1979-07-09T22:29:51TDB.

Notice that while the arrival reference is a full-blown set of orbital elements, we did not use these. Instead, we use the Lambert/Gauss targeting method to plot a course which departs from the center of the Earth on the indicated time, considers only the gravity of the Sun and ignores the gravity of the Earth, and arrives at the center of Jupiter at the indicated time. There is one unique trajectory which crosses the indicated places at the indicated times and is prograde.

So, even though Voyager 1 is launched later, it arrives first. This is for two reasons:

1. Voyager 1 has a higher heliocentric energy
2. Voyager 1 is launched on the "outside track". Normally the inside track is faster, but not in this case. If both spacecraft are going to intercept Jupiter, the first to arrive must be on the outside track because Jupiter is moving from "outside" to "inside".

We end up with the following picture:

There is a lot to look at in this picture, so here are the thousand words. Red is Voyager 1, blue is Voyager 2. The small white circle is the orbit of Earth, with the sun marked as a yellow dot, Jupiter the next larger white circle, and Saturn the largest. The tick marks are 30-day intervals starting at the launch date of Voyager 2. This means that the ticks for Voyager 1 and 2 are directly comparable. We see Voyager 1 pass at about the 4th tick mark. A zoom in reveals that from this perspective directly from Ecliptic North, the trajectories actually cross twice. Normally the trajectory diagrams don't show them crossing at all.

Another interesting thing is that the orbit of Voyager 2 if it did not intercept Jupiter, would have continued about 1AU past Jupiter's orbit. However, Voyager 1's orbit would have gotten almost out to Saturn. Did they use a larger launch vehicle for Voyager 1? No, both spacecraft had almost identical launch speeds (vinf=10.951km/s for Voyager 1, 10.316km/s for Voyager 2). The main difference is just the direction of launch. Voyager 2 was launched above the ecliptic plane (departure asymtote 17 deg above the ecliptic) while Voyager 1 was launched almost in the ecliptic (departure asymtote 4 deg above ecliptic). Also, Voyager 1 was launched closer to the direction of travel of the Earth. Voyager 2 was launched 18 degrees inward, while Voyager 1 was launched 4 deg outward.

Another blast from the past

Once upon a time, I participated in the Internet Ray-tracing Competition. It turns out that they have an archive, and it turns out that most of the source code for my images are still there. I was able to recover another one today:

A modern render of an ancient image

My notes say that this took 6 hours to model and 1h30m45s to render on an AMD K6-2 300MHz 192MiB memory machine. The re-render at the same resolution takes 4.423 seconds on an Intel Xeon E3-1505M laptop with 32GiB memory. This is over 1200x faster. At HD resolution it takes about 13.7 seconds.

Falcon Heavy

Today is the scheduled launch date for the Falcon Heavy demonstration. As of this writing, the launch is on schedule for an 11:30 MST launch.

There is very little official guidance from SpaceX as to what to expect. Elon Musk has stated that minimum mission success is clearing pad 39A far enough such that any further failure doesn't destroy that pad. There has never been a catastrophic failure at pad 39A, and Musk would like to keep it that way.

However, the plan is to do a boost, then three burns of the upper stage. The first finishes launch to LEO. The second is about 30 seconds, and seems to put the booster into a GTO-like orbit. Lifting a heavier spacecraft into full GTO takes about a minute, so there is some hint that this will go into an elliptical orbit that is short of GTO. Part of the demonstration is a 6-hour coast. They are doing it on this flight because the upper stage is very similar to any normal Falcon 9 upper stage, and any demonstration on this stage would apply there. This coast is what would be needed for a 3-burn GSO direct insertion, that apparently is very interesting to the military. For one thing, it would demonstrate that the upper stage could put a GPS satellite directly into its target orbit, like the much more expensive Delta IV medium. A bit more oomph and a similar endurance would put a spacecraft directly into GSO.

In any case, the consensus on NasaSpaceflight is that the target high orbit is one with a period of around 6 hours. After this coast, the second stage would be back at perigee, ready to take maximum advantage of the Oberth effect.

SpaceX has claimed that they will put the payload (A cherry-red Tesla Roadster) into an "earth-mars heliocentric orbit". The launch window for Mars is in May, so they will be launching 3 months out of the window, but since this is such a light payload, they should have plenty of C3 and probably could target Mars if they wanted. However, I think that they will instead target an orbit with periapse at Earth and a C3 typical of launching to Mars. The payload will reach the vicinity of Mars orbit, but Mars will be far far away by then. In fact, to be responsible about Planetary Protection, they should launch into an orbit which will not actually intersect Mars orbit at all, so that there is never any possibility of the car impacting Mars.

Running the numbers based on the Trajectory Planner 1.1.1 from Orbit Hangar, I get a departure C3 of 23.9 km^2/s^2, with a departure today and an arrival on October 17, 2018. The flight time is 252 days. This C3 is high for a Mars launch, but should be doable with such a light payload.

If they are targeting Mars, then the launch vehicle must be able to adjust azimuth in order to target Mars at any time during the window. If they are just going for a given C3, they can use the same azimuth whenever they go. Since ASDS is parked somewhere definite to catch the core stage, I estimate that they are targeting a fixed azimuth independent of launch time.

There are no signs of high-gain antennas or solar panels on the payload, so it is almost certain that once the battery runs down, the payload will become inert. However, the payload is an electric car, with many many amp-hours of battery life. The car radio might run for hours or days.