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Answer from Josh Boehm, the former SpaceX employee:
Will there be an internet connection for those who are going to Mars? Yes and no. Once the infrastructure allows for it, I’m sure there will be an internet connection set up. NASA actually had plans to bring ‘the internet’ to Mars by 2010 with the Mars Telecommunications Orbiter before the project was canceled in 2005. The Mars internet won’t be exactly like the one you know and love here on Earth, though. You could make a second internet on Mars that would work the same way, but it would function independently of Earth’s, and any connection to Earth’s internet (or any other network on Earth) would be painfully slow. How slow, you ask? So slow that you’d finish reading this answer well before getting a single ping response back. Seriously, even if you’re a slow reader. We’ll come back to that.
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The bottleneck of an interplanetary network will be the speed of light, which travels at approximately 299,792 km per second (~186,000 miles per second). Since the distances between planetary bodies are so vast, though, that can seem surprisingly slow. Before we go all the way to Mars, let’s take a look at the Moon.
It takes light (on average) 1.26 seconds to go from the Earth to the Moon. This is just part of the journey, too; a network connection requires a call and response. Every request/response, ping/pong, etc., would be double that, or ~2.52 seconds. It might be usable for basic browsing, but any gamer will tell you that’s awful ping/lag time.
Now I said on average earlier because orbits aren’t perfect circles; they’re ellipses. The minimum and maximum distances from Earth to Mars are about 54.6 million km and about 401 million km. The average is about 225 million km.
So if you do the math, you get a minimum of ~182 seconds (just over 3 minutes), a maximum of ~1,342 seconds (just over 22 minutes), and an average of ~751 seconds (just over 12.5 minutes). This is just for each leg of the journey too, so take those numbers and double them. So that’ll be a range of ~6-44 minutes for your ping time on Mars, depending on where in orbit it is.
This post is about 500 words total, and since the average reader reads approximately 180 words per minute on a monitor, you’d have plenty of time left over to twiddle your thumbs while waiting for the next set of packets to come in. (I told you we’d come back to that.)
So essentially, you’ll be able to download and cache pages for later, but anything interactive or requiring a time-sensitive response from the server or client would not work well. If you’ve ever noticed the lag on a news program when someone is calling in via satellite, that’s just going to Low Earth Orbit and back. Communicating from Mars to Earth, you could record video messages and voicemails for people, but it would be impossible to have a conversation in real-time with current technology.