Iyfjky

Question

Introduction

See background information here.

1000 years after darkness fell on Old Earth, humanity is scattered throughout the solar system. Apart from the Harmonious Republic of Mars, there are colonies in orbit of Venus, Earth and its moon, throughout the asteroid belt, and scattered among the many moons and moonlets and trojans of the Jupiter and Saturn systems.

Problem

As per the rules here, computers are outlawed everywhere in the solar system. In short, the characteristics that make something a computer are electronic memory and being re-programmable.

But, interplanetary trade goes on, even without computers. Powered by nuclear salt-water rockets, torch ships can reach speeds of 100 km/s. Earth to Mars can be done in a few weeks, Jupiter in a few months, and even Saturn in less than a year.

In order to fire these immense engines precisely, you need a clock that is capable of timing the 'burn'. Furthermore, to successfully navigate the solar system, you will need good timekeeping to determine where you are in your orbit in relation to the other planets.

Question

How do you design the 'best' shipboard clock for interplanetary travel, in a future where there are no computers.

Considerations:

The 'best' clock means the most accurate

The clock must be able to be accurately corrected for relativistic effects. 100 km/s isn't that fast, but errors can add up over time.

The clock must give an analog signal output that can be integrated into other electro-mechanical devices. For example, the only way for an old-timey pocket watch to control a burn time is through a human operating a switch.

can i suggest you change the description of a computer from "being programmable." to being "re-programmable" there is an important distinction, programmable systems have been around for centuries, whereas re-programmable systems have only been around for 70 or so years — 8 hours ago
Did you really mean to write "solar systems" (plural) in your introductory paragraph? — 4 hours ago
There is no realistic way to fly starships without computers. The amount of math that is required to process an interplanetary trajectory is mind-boggling. Even if you allow electronic calculators, you still need an equivalent of Space Center Houston, with rooms full of people to support just an individual flight. Maybe you should go the way of Dune, with Mentats and Guild Navigators. — 4 hours ago
@kingledion @ plus tab (maybe a few times) should do it. Has worked fine for me with other people with similar display names. — 3 hours ago
Your title question and body question don't match. My first impression was a clock between worlds, which has been asked before. What you're looking for is, in essence, a navigational clock suitable for the demands of interplanetary travel. May I recommend an edit to match the two questions? — 42 mins ago

score 14 · Answer 1 · 2018-11-06 13:04:17Z

They could use pulsars emission as clock

Neutron stars are very dense, and have short, regular rotational periods. This produces a very precise interval between pulses that ranges from milliseconds to seconds for an individual pulsar. [...] Certain types of pulsars rival atomic clocks in their accuracy in keeping time.

A suitable observatory has to be pointed to one or more pulsars and the collected EM emission is converted to an output electric signal (think something like a photodiode), and by collecting and measuring their emission act as "pace maker" for the ship. By measuring the interval dilation for a pulsar, one can also account for relativistic deviations.

Since rotation is used to stabilize objects in space, the ships could be put in rotation with the rotation axis pointing at a pulsar, providing also an indirect way to measure proper alignment (when signal is lost orientation has been disrupeted)

First, how would you make a clock that is constantly making observations of pulsars without a computer? Second, this doesn't address requirement 3, that it must be able to output an analog electrical signal. — 9 hours ago
@kingledion electromechanical systems got pretty (nay, really) sophisticated in WW2. I'm sure a clever mechanical engineer could figure something out in concert with an electrical engineer. — 9 hours ago
@RonJohn That is more of what I am interested in. I think a quartz timepiece would be sufficient, I just want to know how to turn it into useful equipment without computers. — 8 hours ago
You could handily use a photoresistor and a circuit with known electrical properties, the tricky part is focusing enough light and keeping it on target, but again that should handily be doable with solid state electronics. — 8 hours ago
It's worth noting that such a system would show a slightly different time depending on where you where in relation to the pulsar, and so would be untenable for incredibly long & precise timings if your moving towards or away from the pulsar. — 4 hours ago

score 14 · Answer 2 · 2018-11-06 17:52:59Z

This is probably more of a loophole than the answer you are looking for, but an electronic circuit with a quartz oscillator need not be programmable. It doesn't even need an electronic memory. You'd need to add some frequency dividers, which can also be analog, to provide the final output signal that controls the rocket.

Here's another thought: ultra-precise timing of the burn is only needed if your rocket is on/off only. If you can throttle it back, then you can do the main acceleration/deceleration burn at full power, followed by an orbital correction at much lower power. Or you can use a different set of engines for that part of the manoeuvre.

Addendum

Zubrin's original article says:

In order to be able to analyze the NSWR completely, the
requirement is a computer code which solves the coupled multigroup
neutron transport equation, hydrodynamic equation of
motion, and heat transfer equation as a single system.

So, developing this system without computers is a no-no. However, <Jedi hand gesture> this is not the plot hole you're looking for. In any case, throttling a NSWR seems to be impossible, since the geometry of the plenum chamber must be tuned to the propellant velocity, concentration and yield, ultimately thrust, while remaining strong enough to support a continuous nuclear explosion. You just can't make it telescoping.

The answer then is two engines. The NSWR primaries fire for a fixed amount of time, and and push the vehicle from planetary orbit to transfer The amount of propellant you've used is on the order of 10% of ship's weight. Keep in mind that it might be easier to precisely control the amount of propellant used than the time of the burn.

Then you coast for a few days, carefully observe your trajectory and work out the necessary course corrections. Without computers you give yourself plenty of time and double-check your decimal points! Then fire up your low thrust ion engines an keep them on for hours or days, until you are on track.

score 6 · Answer 3 · 2018-11-06 17:48:20Z

Objective time is irrelevant, only subjective time matters

All your navigation calculations can be done on subjective time. All your shift patterns are going to work on subjective time.

You're far better off building a really good shipboard clock and doing good calculations relative to that, than attempting to automatically convert from some sort of external time prompt. Once you're out of contact with others, the only time that matters is ship time. So for these purposes, we're not going to correct for relativistic effects at all.

So what options are there?

For a traditional no computers environment, we need a good old fashioned mechanical clock. We can cheat and put an electric motor in the back so you don't need to worry about winding it, but we're better off rigging the motor against some nice constant force springs. That way rather than powering the clock itself the motor powers the winding mechanism. Of course this isn't going to be the most accurate option, falling at around 99.9977%

For a more accurate modern approach you want quartz. Something like the Casio F91W, the terrorists favourite bomb trigger due to low cost and legendary accuracy (99.9998%). Again it's a tried and tested system, the accuracy and time loss is a known quantity that can be calculated against. (For a premium system you can get up to 99.9999% accuracy, but where's the fun in that?)

However the trouble with both quartz digital watches or something like an atomic clock is the computer memory, a digital clock is fundamentally a single function computer. And in either case to know when to trigger or end your burns you need to be able to program a time into it to either count up or down to.

However it's possible to build an analogue clock with a quartz timekeeper* giving you a mechanical memory and mechanical output triggers. This should create your "ideal" shipboard clock without upsetting any of the computer zealots. The key difference being the amount of margin for error you're required to put into your navigation calculations at every stage.

Why doesn't absolute time matter?

In the simplest case, because you're going from "here" to "there". When you get "there" you can pick up your new vectors from the local space traffic control. The time where you came from is irrelevant, only the relative position of where you're going next. Your navigation clock doesn't even need to be a clock, it only needs to be a timer. The only times that matter for navigation are the time to next burn and the burn duration.

But we're an exploration vessel, we don't go to ports!

Now it helps if you have a separate clock for your mission time, but it's still not strictly required. You know where you are, you know the location of this place relative to the other places on your route and the calculations for their relative movements. It can all be calculated, it's going to get harder for every extra step you take as the margins for error increase, but it can be done.

This can be made easier with something that will make the over 70s happy. Lookup tables. When you're at point A and visible object B is on vector AB, visible object C is on vector AC, then you can look up your next journey AD, by knowing the difference between AB and AC.

* I won't quote accuracy on these clocks for now as they tend to be at the very cheap end of the market. I'm sure they could be made accurate but there's no call for it.

@kingledion Why would the clock need to account for relativity? You don't need to know the calendar date and time of e.g. your transfer orbit burns, you just need to know your mission time, as experienced by the ship. Any relativistic effects could be far more easily calculated ahead of time and incorporated into the flight plan before launch. — 6 hours ago
@kingledion I don't know enough about clocks to really make a stab at answering the overall question. As I understand relativity though, your perceived time and your acceleration should both appear normal, so relativity wouldn't affect the timing of your burns from your perspective. All it would affect is the date that you arrive at your destination, which you could account for ahead of time. (And being a few days off probably isn't fatal. You'll need to enter orbit and course correct anyway.) — 5 hours ago
You need an absolute time clock to know where planets are going to be later in your journey. The alternative is to look at them constantly, which is a big pain. — 5 hours ago

mlk 1,068310 · Answer 4 · 2018-11-06 16:57:41Z

I'm going to be a bit contrarian and say that any old mechanical pocket-watch will do. It is rather more important to know where you are and how fast you are going. (Which can be solved with radar, radio beacons and other WWII-technology) If you have a torch-ship, you don't need precise clocks at all. At the speeds you are going, we are talking about burn times of at least hours, if not permanent burns in the classic torch-ship style. Burning for a few seconds shorter or longer will not make much of a difference. In fact I guess that impurities in your fuel and slight imperfections in you combustion chamber would have a greater impact. Also, if you are not going quite at the velocity you want to be going, you can just do some adjustment burns, assuming that you left the time for them in your flight plan.

There is also not much need to worry about orbital elements, since the trajectory of a torch ship is really close to a straight line. You just aim at where you want to be going (say the point where Mars is going to be in two weeks), signal "full speed ahead" to your engine-room and then turn around at the midpoint. (Or rather a short bit before, just to be safe.) When you get close enough, you then simply start matching speed until you end up on a slow approach. Getting into a proper orbit in the end might be a bit more finicky, but again, if you have the fuel to spare, it's not terribly hard. See also this nice video on the topic. (And note that he indeed eyeballs the timings and orbital insertion)

All this could actually make the setting more interesting. While any officer who passed his interplanetary navigations class can safely fly a ship, an experienced captain could potentially shave a few hours off the travel time, or a young hotshot lieutenant could ignore the regulation safety margins and go for a suicide burn in the hope of impressing his superiors.

"assuming that you left the time for them in your flight plan." Don't assume. Leave time in your flight plan first, and then calculate the rest of it. Otherwise accidents will happen. — 4 hours ago

ShadoCat 14.3k1851 · Answer 5 · 2018-11-06 19:14:27Z

Allow only mechanical computers. There were difference engines (the first computers) that were purely mechanical.

In Abraham Lincoln's time, the census was performed on punch cards (male/female, age, region, etc.). Mechanical counters advanced each time the proper hole or set of holes were present.

There are also specially carved cams that can open and close valves at predefined, changing rates.

Also, as @Separatrix mentioned, lookup tables work as well.

Space travel isn't impossible without computers, it just takes longer.

You can't go directly from point A to point B. You go to a point near where B should be, stop, take your readings, correct your course and go to another point closer to B, rinse and repeat until you can make it in one more hop.

In this scenario, the better the charts and tables you have and the more skill the pilot has, the closer to B you can get without getting into trouble. This will bring back the ace pilot.

Standard charts and book learning will allow a pilot to make a safe trip from point A to point B but an ace pilot with top of the line charts and equipment (the cams are blueprinted) can shave days or weeks off of the journey because they will need fewer recalculations.

My mother's college dictionary defined computer as: "One who computes."

So break out your slide rules, it's time for some old school SF.

BTW, you should look into George O. Smith's The Complete Venus Equilateral (wiki). This is the author that invented the Star Trek transporters (and had them ruin the economy). Thrust was provided by vacuum tubes and all calculations were done by hand or through machined cams.

jamesqf 9,10211734 · Answer 6 · 2018-11-06 22:04:42Z

The planets themselves, and their moons, form their own clock. Jupiter's moons were in fact used historically, before the invention of accurate chronometers: http://www.oceannavigator.com/May-June-2004/Longitude-by-Jove-navigating-with-Jupiters-moons/

How do you use that on a spaceship without a computer?
â€“Â kingledion
4 mins ago — 4 mins ago

Blade Wraith 7,60311240 · Answer 7 · 2018-11-06 13:21:54Z

Realistically i can't think of a way to do this without a computer... a very complex mechanical system could in theory keep a match of all those different timezones on each different planet or moon, and then keep track of them based off a pendulum swing, but these systems will eventually go out of sync and lose time, keeping time accurately is a very difficult thing to do, that's why we have atomic clocks:

https://www.livescience.com/32660-how-does-an-atomic-clock-work.html

However

why were the computer's banned, presumably because they caused the downfall of Earth, but a multi-planet species is unlikely to ban computers, so there is only one real thing that could be used as a reason to band the survivors together and have them agree to ban computers... Religion

And while i don't want to start any form of religious debate, it is entirely plausible that the figureheads of this new religion could possess an Atomic Clock, now while this does rely on computers these days. they could either be Holy Artefacts or just that the religion is massively hypocritical. think of films such as Equilibium,

where emotions are banned, and that "religion" has clerics that go around and find and kill those that feel and destroy anything that could entcite emotion such as art. and yet the heads of state actually have lavish art decorating the walls etc.

You could easily expand they storyline in this way.

A better answer would discuss how to make an atomic clock without a computer. Is it possible? — 8 hours ago
@kingledion Perhaps, however as i can't see how that would work even slightly and maintain accuracy, i feel it would be completely wrong to even hypothesize about it — 7 hours ago
@kingledion You might need a computer for an atomic clock (I don't know), but do you need a reprogrammable computer for that? — 4 hours ago

score 14 · Answer 8 · 2018-11-06 13:04:17Z

They could use pulsars emission as clock

Neutron stars are very dense, and have short, regular rotational periods. This produces a very precise interval between pulses that ranges from milliseconds to seconds for an individual pulsar. [...] Certain types of pulsars rival atomic clocks in their accuracy in keeping time.

A suitable observatory has to be pointed to one or more pulsars and the collected EM emission is converted to an output electric signal (think something like a photodiode), and by collecting and measuring their emission act as "pace maker" for the ship. By measuring the interval dilation for a pulsar, one can also account for relativistic deviations.

Since rotation is used to stabilize objects in space, the ships could be put in rotation with the rotation axis pointing at a pulsar, providing also an indirect way to measure proper alignment (when signal is lost orientation has been disrupeted)

First, how would you make a clock that is constantly making observations of pulsars without a computer? Second, this doesn't address requirement 3, that it must be able to output an analog electrical signal. — 9 hours ago
@kingledion electromechanical systems got pretty (nay, really) sophisticated in WW2. I'm sure a clever mechanical engineer could figure something out in concert with an electrical engineer. — 9 hours ago
@RonJohn That is more of what I am interested in. I think a quartz timepiece would be sufficient, I just want to know how to turn it into useful equipment without computers. — 8 hours ago
You could handily use a photoresistor and a circuit with known electrical properties, the tricky part is focusing enough light and keeping it on target, but again that should handily be doable with solid state electronics. — 8 hours ago
It's worth noting that such a system would show a slightly different time depending on where you where in relation to the pulsar, and so would be untenable for incredibly long & precise timings if your moving towards or away from the pulsar. — 4 hours ago

score 14 · Answer 9 · 2018-11-06 17:52:59Z

This is probably more of a loophole than the answer you are looking for, but an electronic circuit with a quartz oscillator need not be programmable. It doesn't even need an electronic memory. You'd need to add some frequency dividers, which can also be analog, to provide the final output signal that controls the rocket.

Here's another thought: ultra-precise timing of the burn is only needed if your rocket is on/off only. If you can throttle it back, then you can do the main acceleration/deceleration burn at full power, followed by an orbital correction at much lower power. Or you can use a different set of engines for that part of the manoeuvre.

Addendum

Zubrin's original article says:

In order to be able to analyze the NSWR completely, the
requirement is a computer code which solves the coupled multigroup
neutron transport equation, hydrodynamic equation of
motion, and heat transfer equation as a single system.

So, developing this system without computers is a no-no. However, <Jedi hand gesture> this is not the plot hole you're looking for. In any case, throttling a NSWR seems to be impossible, since the geometry of the plenum chamber must be tuned to the propellant velocity, concentration and yield, ultimately thrust, while remaining strong enough to support a continuous nuclear explosion. You just can't make it telescoping.

The answer then is two engines. The NSWR primaries fire for a fixed amount of time, and and push the vehicle from planetary orbit to transfer The amount of propellant you've used is on the order of 10% of ship's weight. Keep in mind that it might be easier to precisely control the amount of propellant used than the time of the burn.

Then you coast for a few days, carefully observe your trajectory and work out the necessary course corrections. Without computers you give yourself plenty of time and double-check your decimal points! Then fire up your low thrust ion engines an keep them on for hours or days, until you are on track.

score 6 · Answer 10 · 2018-11-06 17:48:20Z

Objective time is irrelevant, only subjective time matters

All your navigation calculations can be done on subjective time. All your shift patterns are going to work on subjective time.

You're far better off building a really good shipboard clock and doing good calculations relative to that, than attempting to automatically convert from some sort of external time prompt. Once you're out of contact with others, the only time that matters is ship time. So for these purposes, we're not going to correct for relativistic effects at all.

So what options are there?

For a traditional no computers environment, we need a good old fashioned mechanical clock. We can cheat and put an electric motor in the back so you don't need to worry about winding it, but we're better off rigging the motor against some nice constant force springs. That way rather than powering the clock itself the motor powers the winding mechanism. Of course this isn't going to be the most accurate option, falling at around 99.9977%

For a more accurate modern approach you want quartz. Something like the Casio F91W, the terrorists favourite bomb trigger due to low cost and legendary accuracy (99.9998%). Again it's a tried and tested system, the accuracy and time loss is a known quantity that can be calculated against. (For a premium system you can get up to 99.9999% accuracy, but where's the fun in that?)

However the trouble with both quartz digital watches or something like an atomic clock is the computer memory, a digital clock is fundamentally a single function computer. And in either case to know when to trigger or end your burns you need to be able to program a time into it to either count up or down to.

However it's possible to build an analogue clock with a quartz timekeeper* giving you a mechanical memory and mechanical output triggers. This should create your "ideal" shipboard clock without upsetting any of the computer zealots. The key difference being the amount of margin for error you're required to put into your navigation calculations at every stage.

Why doesn't absolute time matter?

In the simplest case, because you're going from "here" to "there". When you get "there" you can pick up your new vectors from the local space traffic control. The time where you came from is irrelevant, only the relative position of where you're going next. Your navigation clock doesn't even need to be a clock, it only needs to be a timer. The only times that matter for navigation are the time to next burn and the burn duration.

But we're an exploration vessel, we don't go to ports!

Now it helps if you have a separate clock for your mission time, but it's still not strictly required. You know where you are, you know the location of this place relative to the other places on your route and the calculations for their relative movements. It can all be calculated, it's going to get harder for every extra step you take as the margins for error increase, but it can be done.

This can be made easier with something that will make the over 70s happy. Lookup tables. When you're at point A and visible object B is on vector AB, visible object C is on vector AC, then you can look up your next journey AD, by knowing the difference between AB and AC.

* I won't quote accuracy on these clocks for now as they tend to be at the very cheap end of the market. I'm sure they could be made accurate but there's no call for it.

@kingledion Why would the clock need to account for relativity? You don't need to know the calendar date and time of e.g. your transfer orbit burns, you just need to know your mission time, as experienced by the ship. Any relativistic effects could be far more easily calculated ahead of time and incorporated into the flight plan before launch. — 6 hours ago
@kingledion I don't know enough about clocks to really make a stab at answering the overall question. As I understand relativity though, your perceived time and your acceleration should both appear normal, so relativity wouldn't affect the timing of your burns from your perspective. All it would affect is the date that you arrive at your destination, which you could account for ahead of time. (And being a few days off probably isn't fatal. You'll need to enter orbit and course correct anyway.) — 5 hours ago
You need an absolute time clock to know where planets are going to be later in your journey. The alternative is to look at them constantly, which is a big pain. — 5 hours ago

mlk 1,068310 · Answer 11 · 2018-11-06 16:57:41Z

I'm going to be a bit contrarian and say that any old mechanical pocket-watch will do. It is rather more important to know where you are and how fast you are going. (Which can be solved with radar, radio beacons and other WWII-technology) If you have a torch-ship, you don't need precise clocks at all. At the speeds you are going, we are talking about burn times of at least hours, if not permanent burns in the classic torch-ship style. Burning for a few seconds shorter or longer will not make much of a difference. In fact I guess that impurities in your fuel and slight imperfections in you combustion chamber would have a greater impact. Also, if you are not going quite at the velocity you want to be going, you can just do some adjustment burns, assuming that you left the time for them in your flight plan.

There is also not much need to worry about orbital elements, since the trajectory of a torch ship is really close to a straight line. You just aim at where you want to be going (say the point where Mars is going to be in two weeks), signal "full speed ahead" to your engine-room and then turn around at the midpoint. (Or rather a short bit before, just to be safe.) When you get close enough, you then simply start matching speed until you end up on a slow approach. Getting into a proper orbit in the end might be a bit more finicky, but again, if you have the fuel to spare, it's not terribly hard. See also this nice video on the topic. (And note that he indeed eyeballs the timings and orbital insertion)

All this could actually make the setting more interesting. While any officer who passed his interplanetary navigations class can safely fly a ship, an experienced captain could potentially shave a few hours off the travel time, or a young hotshot lieutenant could ignore the regulation safety margins and go for a suicide burn in the hope of impressing his superiors.

"assuming that you left the time for them in your flight plan." Don't assume. Leave time in your flight plan first, and then calculate the rest of it. Otherwise accidents will happen. — 4 hours ago

ShadoCat 14.3k1851 · Answer 12 · 2018-11-06 19:14:27Z

Allow only mechanical computers. There were difference engines (the first computers) that were purely mechanical.

In Abraham Lincoln's time, the census was performed on punch cards (male/female, age, region, etc.). Mechanical counters advanced each time the proper hole or set of holes were present.

There are also specially carved cams that can open and close valves at predefined, changing rates.

Also, as @Separatrix mentioned, lookup tables work as well.

Space travel isn't impossible without computers, it just takes longer.

You can't go directly from point A to point B. You go to a point near where B should be, stop, take your readings, correct your course and go to another point closer to B, rinse and repeat until you can make it in one more hop.

In this scenario, the better the charts and tables you have and the more skill the pilot has, the closer to B you can get without getting into trouble. This will bring back the ace pilot.

Standard charts and book learning will allow a pilot to make a safe trip from point A to point B but an ace pilot with top of the line charts and equipment (the cams are blueprinted) can shave days or weeks off of the journey because they will need fewer recalculations.

My mother's college dictionary defined computer as: "One who computes."

So break out your slide rules, it's time for some old school SF.

BTW, you should look into George O. Smith's The Complete Venus Equilateral (wiki). This is the author that invented the Star Trek transporters (and had them ruin the economy). Thrust was provided by vacuum tubes and all calculations were done by hand or through machined cams.

jamesqf 9,10211734 · Answer 13 · 2018-11-06 22:04:42Z

The planets themselves, and their moons, form their own clock. Jupiter's moons were in fact used historically, before the invention of accurate chronometers: http://www.oceannavigator.com/May-June-2004/Longitude-by-Jove-navigating-with-Jupiters-moons/

How do you use that on a spaceship without a computer?
â€“Â kingledion
4 mins ago — 4 mins ago

Blade Wraith 7,60311240 · Answer 14 · 2018-11-06 13:21:54Z

Realistically i can't think of a way to do this without a computer... a very complex mechanical system could in theory keep a match of all those different timezones on each different planet or moon, and then keep track of them based off a pendulum swing, but these systems will eventually go out of sync and lose time, keeping time accurately is a very difficult thing to do, that's why we have atomic clocks:

https://www.livescience.com/32660-how-does-an-atomic-clock-work.html

However

why were the computer's banned, presumably because they caused the downfall of Earth, but a multi-planet species is unlikely to ban computers, so there is only one real thing that could be used as a reason to band the survivors together and have them agree to ban computers... Religion

And while i don't want to start any form of religious debate, it is entirely plausible that the figureheads of this new religion could possess an Atomic Clock, now while this does rely on computers these days. they could either be Holy Artefacts or just that the religion is massively hypocritical. think of films such as Equilibium,

where emotions are banned, and that "religion" has clerics that go around and find and kill those that feel and destroy anything that could entcite emotion such as art. and yet the heads of state actually have lavish art decorating the walls etc.

You could easily expand they storyline in this way.

A better answer would discuss how to make an atomic clock without a computer. Is it possible? — 8 hours ago
@kingledion Perhaps, however as i can't see how that would work even slightly and maintain accuracy, i feel it would be completely wrong to even hypothesize about it — 7 hours ago
@kingledion You might need a computer for an atomic clock (I don't know), but do you need a reprogrammable computer for that? — 4 hours ago

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