## Introduction

I was reading an article about the Voyager 1 space craft nearing the edge of interstellar space. This article was so interesting that I ended up reading a number of articles on the subject (one example) and they all had the same numbers in them:

- Voyager 1 is leaving our solar system at 3.6 Astronomical Units (AU) per year.
- Voyager 1 is moving toward the star Gliese 445 (aka AC+79 3888) and will be at the point of closest approach in about 40,000 years
- The closest Gliese 445 and Voyager 1 will get is about 1.7 light-years

My interest in these numbers was raised when I noticed that Voyager 1's velocity is too low for it to come near Gliese 445 in 40,000 years. It turned out that Gliese 445 is coming to us faster than Voyager is going to it. I was a bit surprised.

Let's see if we can gain some insight into these numbers. I do not want to get into the details of astrometry, so my work will be approximate (i.e. Fermi problem analysis).

## Background

The following background material is useful to review before we dive into the numbers.

### Voyager 1

Voyager 1 was a space probe launched in 1977 and it is still performing scientific work. It is currently heading out the solar system. Figure 1 shows its path (labeled V1) relative to the orbital plane (Source).

After a very long time, Voyager 1 will pass relatively close to Gliese 445. This post will estimate that time and distance.

### Gliese Star Catalog

If you are curious how stars get names like this, check out the Wikipedia entry on the Gliese Star Catalog. This star catalogue lists the stars located within 25 parsecs (81.54 light-year) of the Earth.

### Star Distances Versus Time

Figure 2 shows a graph of the distances of various nearby stars to Earth as a function of time (Source).

Observe how much the stars move over long time periods. Note also how Gliese 445 is moving towards the Earth, which means it is also coming closer to Voyager 1. The shape of the distance curve is described by a hyperbola. We derive this curve shape as shown in Figure 3.

## Analysis

We are going to compute a few interesting numbers:

- What is the velocity of Voyager 1 in AU per year?
- How long does Voyager 1 take to travel one light year?
- How fast is Gliese 445 approaching the Earth?
- When will Voyager 1 have its closest approach to Gliese 445?
- How close will Voyager 1 get to Gliese 445?

### Calculations

#### Voyage 1 Velocity in AU Per Year and the Time for It to Travel One Light Year

Figure 4 shows my calculations for the velocity of Voyager 1 in AU per year (3.595 AU/year) and how long it takes for Voyager 1 to travel one light-year (17,600 years). These numbers agree with those in this NASA article.

#### Gliese 445 Approach Speed to Earth and Time of Closest Approach

I digitized the distance curve for Gliese 445 in Figure 2 and took the derivative of position versus time. Since the graph for Gliese 445 does not show data for today, I will compute the speed for 20,000 years from now. The distance versus time curve looks fairly linear in this region and the velocity should be similar to what we see today (Figure 5).

I compute an approach velocity of 106 km/sec (much higher than Voyager 1's velocity of 17 km/sec). The Wikipedia lists this value as 119 km/sec -- close enough. I compute that the time of closest approach is 46,000 years from now, which is close to the Wikipedia statement of "about 40,000 years". I compute the distance at closest approach as 3.485 light-years, which is close to the Wikipedia value of 3.45 light-years.

#### Closest Distance of Voyager 1 to Gliese 445

To get an accurate estimate of this distance, I would need to know details on the exact relationship between the courses of Voyager 1 and Gliese 445, which I do not have access to. To get a rough estimate of how close they could get, let's assume that Voyager 1 is moving toward the point of closest approach between Gliese 445 and the Sun (Figure 6). This will give us an answer that is smaller than the true value, but it will be close enough for my purposes.

My result shows that if Voyager was perfectly aimed at Gliese 445, it would get within 0.9 light-years of the star. NASA reports that the distance at closest approach is 1.7 light-years, a number for which I assume they took into account the actual aiming error. My simple result is not too far off.

## Conclusion

Just a quick exercise to make sure that I understand what I am reading. The idea that Gliese 445 is approaching the Sun so quickly was the news to me!

The illustrated curves look parabolic---never thought about it that way! Great.

I was a bit surprise about the shape myself until I thought about it. If we assume that the Sun is fixed and Gliese 445 is moving on a straight line by us, the distance between the Sun and Gliese 445 must be hyperbolic with time.

For readers that I have not done this work for awhile, I have added a figure with derivation on the topic to the blog post.

Thank you for a nice article. 🙂

Just one remark - I believe the distance versus time graph is hyperbolic, not parabolic. Two reasons: a) the curve must converge to a straight line in infinity (i.e. d_min is insignificant when Gliese 445 is very far) and b) it can be converted to hyperbola equation :-).

Many thanks for the correction! I have updated the blog post and included a derivation that shows the distance function is a hyperbola.

Again, thanks.

Mathscinotes

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I found this intriguing as well, I had worked out that it would take Voyager about 80,000 Earth years to reach AC but when I read that a star 17.6LY away would be the first that Voyager would encounter I thought ?????? How does that work? I found that Voyager would pass Gliese 445 in about 40k years when G445 was about 3.46LY from our sun, a few quick calculations revealed the Sun and G445 converging at a relative speed of about 105km/s. I see in the diagrams that AC / AP, Barnards and a number of other stars are making close approaches to the sun over the next 60k years. I knew that stars moved in relation to each other but this just blew me away. Awesome stuff.

Your article was very informative.

Cheers

Greg

As I near retirement, I plan on spending as much time as I can on my amateur astronomy activities. I wish I could write more about astronomy -- I am knee-deep in some hard-core electronic design work right now.

Thank you for the comment.

mathscinotes

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Great job. I liked it a lot