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I remember the Apollo missions and I'm pretty sure it took about three days to reach the moon.

India’s space agency is set to launch its homegrown Chandrayaan-3 moon mission from the southeastern island of Sriharikota, at the Satish Dhawan Space Centre on July 14 at 2:35pm local time (09:05 GMT).

A 43.5-metre (143-feet) Launch Vehicle Mark-III, or LVM3, launch rocket will blast the spacecraft into an elliptical Earth orbit before it loops towards the moon for a scheduled landing near the moon’s south pole around August 23.

Why 40 days instead of 3?

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    $\begingroup$ Since humans needing life support are not on board one can take a more efficient trajectory. $\endgroup$
    – Jon Custer
    Jul 15 at 14:46
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    $\begingroup$ @JonCuster I wonder how much of the extra 37 days is actually necessary for the increase in efficiency, and how much is just conservative mission planning; trajectory and systems checks, corrections, etc. since there's no need to rush due to "perishables on board". $\endgroup$
    – uhoh
    Jul 15 at 15:21
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    $\begingroup$ Their vehicle does not have anywhere near the performance of the Apollo vehicles. $\endgroup$
    – Organic Marble
    Jul 15 at 18:34
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    $\begingroup$ In effect, it is using the earth for gravity assist, to boost its orbit higher and higher. This is much more fuel-effiicient than a direct burn, but takes significantly longer. $\endgroup$
    – TonyK
    Jul 16 at 0:12
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    $\begingroup$ There are many things considered when planning a mission. Hurry is often not one of them. You need to consider the fuel efficiency, the lifetime of the payload, the sun angles at the destination if you are doing photography, where on the moon you want to arrive, etc. So what if you don't have people aboard and it takes a month longer to get a better mission? -1 $\endgroup$
    – Ross Millikan
    Jul 16 at 3:56

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This is the trajectory of its predecessor, Chandrayaan-2:

orbit raising before lunar capture

Due to the low power of the spacecraft engine, it raises its orbit over multiple passes (and makes the most of the Oberth effect).

As the orbit is raised, it takes longer to complete the orbit before the next burn. In addition, there appears to be a number of orbits where no burn takes place. Presumably this is to allow wriggle room to still achieve a rendezvous if an early burn fails (although it might be for more practical reasons, such as communications links, or perhaps even just so they happen during office hours).

I still feel it's worth noting the final trip to the moon "only" takes around six days.

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    $\begingroup$ Final trip takes more like 5-7 days (14thish through 21stish) which is typical for a Hohmann transfer from low Earth orbit to the Moon's distance. Apollo burned more delta-v on departure to cut things down to three. $\endgroup$
    – notovny
    Jul 15 at 21:08
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    $\begingroup$ @TheRocketfan India really doesn't have a 24-7 monitoring/communication/telemetry network so they need extra time to get everything checked and done. When China launched Chang'e 4, they had 5 ships and a dozen ground stations spread across the globe to support it and even launched a relay satellite towards moon's L2 specifically for Chang'e. With such a tight budget, India is doing their moon mission more or less like others doing a Mars mission. $\endgroup$
    – user3528438
    Jul 16 at 3:48
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    $\begingroup$ @user3528438: I would guess that if there's any possible imprecision in any of the burns, it may be difficult to precisely measure the new orbits until the craft has passed the point where one would want the next burn to start. If one measures two nearby passes without any commanded acceleration between them, then it will be possible to know precisely when the engines should be used on the thir orbit. $\endgroup$
    – supercat
    Jul 16 at 18:43
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    $\begingroup$ (to a space newbie) what an incredibly unintuitive trajectory! (I'd have assumed approximately a straight line!). Thanks for making the animated gif! $\endgroup$
    – stevec
    Jul 18 at 0:52
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    $\begingroup$ @stevec Orbital mechanics are a little mindbending until you actually feel them. If I could, I'd like to recommend to you the game "Kerbal Space Program". Once you're comfortable with the Oberth effect as shown in the animation, try launching in an equatorial orbit and shifting to a polar one once in space. That's a real thinker. $\endgroup$
    – Adam Barnes
    Jul 18 at 9:18

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