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The question behind the formula · JEE Physics: Gravitation

Why did Chandrayaan-3 take 40 days to land on the Moon, when Apollo 11 took four?

Apollo 11 crossed to the Moon in three days. Chandrayaan-3 spent seventeen days just going round Earth. The slow route was not a weakness: it is the gravitation chapter of your syllabus, used by a mission with a smaller rocket. Every number below is worked out from ISRO's own orbit figures.

Short answer

Because it climbed instead of leaping. Apollo 11's rocket gave it enough speed to head straight for the Moon. Chandrayaan-3's rocket left it in a long orbit around Earth, and the spacecraft raised that orbit in five steps with its own engine, mostly firing at the low point, where a burn adds the most energy. The crossing itself took under five days; the rest went on climbing away from Earth, settling into orbit around the Moon and waiting for sunrise at the landing site.

By 10 min read
Contents
  1. Where the days went
  2. Why not straight there
  3. Reading an orbit
  4. Burning at the low point
  5. Why each step took days
  6. Slowing down at the Moon
  7. Why 23 August
  8. Practice
Terms used in this essay
Perigee, apogee
The lowest and highest points of an orbit around Earth, given as heights above the surface.
Perilune, apolune
The same two points for an orbit around the Moon.
“170 × 36,500 km”
ISRO's way of writing an orbit: the perigee height and the apogee height. Not a multiplication.
Semi-major axis, \( a \)
The size of an orbit: half the sum of the lowest and highest points' distances from Earth's centre.
Burn
One firing of the spacecraft's engine, lasting minutes, to speed it up or slow it down.
Escape speed
The speed at which a spacecraft would coast away from Earth, or the Moon, and never fall back.
Period
The time for one trip around an orbit.
Orbital energy per kg
Kinetic plus potential energy for each kilogram of spacecraft, in megajoules (MJ). Negative while Earth holds it in orbit; zero at escape.
Trans-lunar injection
The burn on 1 August that sent the spacecraft out to the Moon.
Sphere of influence
The region within about 66,000 km of the Moon, inside which the Moon's pull, rather than Earth's, is what bends a spacecraft's path relative to the Moon. Earth still pulls, but it pulls the Moon and the spacecraft almost equally.
Powered descent
The last 20 minutes of the landing, with the lander's engines braking it from orbital speed to a standstill on the surface.
Propulsion module
The part of Chandrayaan-3 that carried the main engine and fuel from Earth orbit to lunar orbit.
Lander, rover
Vikram, which landed, and Pragyan, the small six-wheeled vehicle it carried.
Geostationary orbit
An orbit with a period of one day, in which a satellite stays over one spot on the equator.
Oberth effect
The extra energy gained by firing an engine where a spacecraft is moving fastest.
SyllabusJEE Main 2026, Unit 6: “Kepler's law of planetary motion … Escape velocity, motion of a satellite, orbital velocity, time period and energy of satellite.” This essay uses all of them.

Chandrayaan-3 lifted off from Sriharikota at 2:35 pm IST on 14 July 2023, on ISRO's heaviest rocket, LVM3. Its lander, Vikram, touched down near the Moon's south pole just after 6 pm IST on 23 August, forty days later. Apollo 11 left Florida on 16 July 1969 and landed on the Moon on 20 July, four days later.

Where did the 40 days go?

Not into the crossing. Lay the two missions side by side and the trip from Earth to the Moon takes about the same time in both: three days for Apollo 11, under five for Chandrayaan-3. The difference is everything either side of it.

Where the 40 days went: Apollo 11 and Chandrayaan-3 Two timelines, in days after launch. Apollo 11, 1969: under three hours in Earth orbit, three days crossing to the Moon, a day in lunar orbit, and a landing after 4.3 days. Chandrayaan-3, 2023: 17.4 days raising its orbit around Earth, 4.8 days crossing, 18 days lowering its orbit around the Moon and waiting for sunrise at the landing site, and a landing after 40 days. The crossings take about as long as each other. Apollo 11 1969 landed after 4.3 days, 3.0 of them crossing Chandrayaan-3 2023 around Earth · 17.4 days crossing · 4.8 days around the Moon · 18.0 days 0 5 10 15 20 25 30 35 40 days after launch · white ticks inside the bar: days with an engine burn Where the 40 days went: Apollo 11 and Chandrayaan-3 Two timelines, in days after launch. Apollo 11, 1969: under three hours in Earth orbit, three days crossing to the Moon, a day in lunar orbit, and a landing after 4.3 days. Chandrayaan-3, 2023: 17.4 days raising its orbit around Earth, 4.8 days crossing, 18 days lowering its orbit around the Moon and waiting for sunrise at the landing site, and a landing after 40 days. The crossings take about as long as each other. Apollo 11 (1969) landed after 4.3 days, 3.0 of them crossing Chandrayaan-3 (2023) around Earth · 17.4 days crossing · 4.8 days around the Moon · 18.0 days 0 10 20 30 40 days after launch · ticks: burn days
Where the days went. Apollo 11 left Earth orbit under three hours after launch. Chandrayaan-3 spent 17 days raising its orbit around Earth and 18 lowering it around the Moon.

The animation below follows all 40 days, to scale. Step through it, or press Play, and watch the date in its corner: the days pile up during the laps around Earth and around the Moon, not on the way between them.

Chandrayaan-3's crossing to the Moon, to scale To scale: the spacecraft's crossing from Earth to the Moon, 1 to 5 August 2023. Earth sits at the lower left inside its earlier, smaller orbits. From its lowest point, just beside Earth, the spacecraft goes up and out along a long, flat orbit to the right, towards the Moon's distance. The Moon travels clockwise along its own orbit, from the top of the drawing down to the same place. Marks for 2, 3 and 4 August on both paths show the spacecraft fast at first and slowing, and the Moon moving steadily, until they meet on 5 August. A dashed circle around the Moon there marks its sphere of influence, about 66,000 kilometres, inside which the spacecraft swings round the Moon before being captured. The crossing, 1–5 August, to scale 2 Aug 2 Aug 3 Aug 3 Aug 4 Aug 4 Aug Earth Moon on 1 August They meet, 5 August Chandrayaan-3: 4.8 days out,slowing all the way The Moon, along its own orbit,1.02 km/s Moon's sphere of influence,66,000 km: inside it, the Moon steers 100,000 km Earth to scale; the Moon drawn slightly larger. Chandrayaan-3's crossing to the Moon, to scale To scale: the spacecraft's crossing from Earth to the Moon, 1 to 5 August 2023. Earth sits at the lower left inside its earlier, smaller orbits. From its lowest point, just beside Earth, the spacecraft goes up and out along a long, flat orbit to the right, towards the Moon's distance. The Moon travels clockwise along its own orbit, from the top of the drawing down to the same place. Marks for 2, 3 and 4 August on both paths show the spacecraft fast at first and slowing, and the Moon moving steadily, until they meet on 5 August. A dashed circle around the Moon there marks its sphere of influence, about 66,000 kilometres, inside which the spacecraft swings round the Moon before being captured. The crossing, 1–5 Aug, to scale 2 Aug 2 Aug 3 Aug 3 Aug 4 Aug 4 Aug Earth Moon, 1 Aug they meet, 5 Aug Chandrayaan-3: 4.8 days out,slowing all the way The Moon,1.02 km/s Moon's sphereof influence 100,000 km Moon drawn slightly larger
  1. 14 July · Launch. The rocket leaves Chandrayaan-3 in an orbit 170 km above Earth at its lowest point and 36,500 km at its highest. Step forward to follow it for 40 days.
  2. 15 July · Burn 1. Two laps of 10.7 hours: fast at the lowest point (10.28 km/s), slow at the highest (1.57 km/s). Then a burn at the lowest point lifts the highest point to 41,762 km.
  3. 17 July · Burn 2. Three and a half laps, then a burn at the highest point, which lifts the lowest point instead: 173 km to 226 km.
  4. 18 July · Burn 3. Two and a half laps, then a burn at the lowest point: the highest point goes out to 51,400 km.
  5. 20 July · Burn 4. Three laps, then a burn at the lowest point: the highest point goes out to 71,351 km.
  6. 25 July · Burn 5. Five laps, then a burn at the lowest point: the highest point goes out to 127,609 km. Each lap now takes longer.
  7. 1 August · Sent to the Moon. Three laps of 52 hours, then the last burn around Earth: the highest point goes out to 369,328 km, as far as the Moon.
  8. 1–5 August · The crossing. It slows all the way out while the Moon, at 1.02 km/s, comes round to meet it. Falling past the Moon it reaches 2.41 km/s; a 1,835-second burn slows it to 2.17 km/s, and the Moon holds it.
  9. 6 August · Lowered. A burn at the lowest point, against the direction of travel, brings the highest point down from 18,074 km to 4,313 km.
  10. 9 August · Lowered. Again at the lowest point: the highest point comes down to 1,437 km.
  11. 14 August · Nearly circular. 150 km up at the lowest point, 177 km at the highest: the speed barely changes now.
  12. 16 August · Circular. Between 153 and 163 km above the Moon all the way round.
  13. 17 August · The lander separates. The lander, Vikram, leaves the propulsion module, which stays in this orbit.
  14. 18 August · The lander alone. Its first burn, at its highest point, brings its lowest point down to 113 km (highest 157 km).
  15. 20 August · Lowered for landing. A second burn: lowest point 25 km, highest point 134 km. The landing will start from the lowest point.
  16. 23 August · Landing. At about 17:44 IST, 25 km up and moving at 1.69 km/s (worked out from the orbit), the engines start braking. At 18:04 the lander is down near 69.4°S, 32.3°E: 40 days after launch.
The whole journey, to scale. Orbit heights, burn dates and the capture and landing times are ISRO's; positions and speeds are worked out from them. Simplified: drawn in one flat plane, and around the Moon about one lap of each orbit is shown, where the spacecraft went round many times.

So the real question is why Chandrayaan-3 had to climb away from Earth in steps, instead of being sent straight on its way.

Why couldn't its rocket send it straight to the Moon?

Perigee and apogeeThe lowest and highest points of an orbit around Earth (around the Moon: perilune and apolune). ISRO writes an orbit as these two heights joined by ×, as in “170 × 36,500 km”. The × is not a multiplication.

Because heading for the Moon from a low orbit takes very nearly escape speed, and a spacecraft's speed is bought with rocket. Escape speed from a distance \( r \) from Earth's centre is

$$ v_{\text{esc}} = \sqrt{\frac{2GM}{r}} $$

In words: the speed at which something at distance \( r \) from Earth's centre would coast away and never fall back. \( G \) is the gravitational constant and \( M \) Earth's mass; their product \( GM \) is 3.986 × 1014 m3/s2.

About 290 km above Earth, escape speed is 10.94 km/s. Apollo 11's Saturn V, the most powerful rocket of its day, put the spacecraft on a path to the Moon two hours and forty-four minutes after launch. LVM3 is a much smaller rocket. It left the 3,895 kg Chandrayaan-3 in a long, stretched orbit: 170 km above Earth at its lowest point and 36,500 km at its highest, moving at 10.28 km/s at the lowest point. That is 0.66 km/s short of escape speed there. The spacecraft had to make up the rest itself, with the engine in its propulsion module (the part that carried the main engine and its fuel), in a series of burns. A burn is one firing of the engine, lasting minutes, that speeds the spacecraft up or slows it down.

How do you read an orbit like “170 × 36,500 km”?

As two heights, not a multiplication. ISRO describes every orbit by its lowest point and its highest point, both measured above Earth's surface, joined by ×. So “170 × 36,500 km” means the spacecraft came within 170 km of the surface at the closest point of each orbit, and went out to 36,500 km above it at the farthest. The lowest point is called the perigee and the highest the apogee. In this essay we write the two heights out in words.

How to read an orbit: lowest point × highest point How to read an orbit written as 170 by 36,500 kilometres. Not to scale. Earth sits at one focus of an ellipse. On the left, the orbit's lowest point, the perigee, is 170 kilometres above Earth's surface. On the right, its highest point, the apogee, is 36,500 kilometres above the surface. Formulas measure from Earth's centre, so add Earth's radius, 6,378 kilometres: the lowest point is 6,548 kilometres from the centre and the highest point 42,878. Half the sum, 24,713 kilometres, is the semi-major axis a, the orbit's size in every formula. How to read “170 × 36,500 km” (the launch orbit) not to scale lowest point(perigee):170 km up highestpoint(apogee) 36,500 km above the surface Earth's radius R = 6,378 km rₚ = 6,378 + 170 = 6,548 km from the centre rₐ = 6,378 + 36,500 = 42,878 km from the centre Orbit size: a = (rₚ + rₐ) ÷ 2 = 24,713 km, the semi-major axis Heights are measured from the surface; formulas need distances from the centre. How to read an orbit: lowest point × highest point How to read an orbit written as 170 by 36,500 kilometres. Not to scale. Earth sits at one focus of an ellipse. On the left, the orbit's lowest point, the perigee, is 170 kilometres above Earth's surface. On the right, its highest point, the apogee, is 36,500 kilometres above the surface. Formulas measure from Earth's centre, so add Earth's radius, 6,378 kilometres: the lowest point is 6,548 kilometres from the centre and the highest point 42,878. Half the sum, 24,713 kilometres, is the semi-major axis a, the orbit's size in every formula. Reading “170 × 36,500 km” not to scale 1 2 3 4 1 lowest point (perigee): 170 km above the surface 2 highest point (apogee): 36,500 km above the surface 3 Earth's radius R = 6,378 km 4 from Earth's centre: rₚ = 6,548 km, rₐ = 42,878 km Orbit size: a = (rₚ + rₐ) ÷ 2 = 24,713 km Heights are measured from the surface;formulas need distances from the centre.
Reading an orbit. The two heights ISRO gives are measured from Earth's surface. The formulas in your textbook measure from Earth's centre, so add Earth's radius, 6,378 km, to each one first. Half the sum of the two distances is \( a \), the size of the orbit.

Every formula below uses three numbers worked out this way: \( r_p \), the lowest point's distance from Earth's centre; \( r_a \), the highest point's; and \( a \), half their sum, called the semi-major axis. For the launch orbit they are 6,548 km, 42,878 km and 24,713 km.

The spacecraft3,895 kg at launch: a propulsion module of about 2,150 kg, and the Vikram lander, about 1,750 kg including the 26 kg Pragyan rover.

How does a burn at the low point raise the high point?

Start with the energy of an orbit. For a body of mass \( m \) on an orbit whose semi-major axis is \( a \), kinetic and potential energy add up to

$$ E = -\frac{GMm}{2a} $$

In words: the energy depends only on the orbit's size, \( a \). It is negative because the spacecraft is still held by Earth; zero would mean just enough to escape. A bigger orbit has a less negative energy, so making the orbit bigger means adding energy.

For the launch orbit (\( a \) = 24,713 km) it comes to −8.06 MJ for every kilogram of spacecraft. MJ is megajoules, millions of joules; dividing by the mass makes the number the same for any spacecraft on that orbit.

Where you add it matters. A burn that raises the speed from \( v \) to \( v + \Delta v \) adds kinetic energy

$$ \Delta E = \tfrac12 m (v + \Delta v)^2 - \tfrac12 m v^2 = m\,v\,\Delta v + \tfrac12 m\,(\Delta v)^2 $$

In words: the same push (\( \Delta v \)) adds more energy when the spacecraft is already moving fast, because of the \( v \) in \( m\,v\,\Delta v \).

So a burn adds the most energy where \( v \) is largest: at the lowest point. On the launch orbit, a 100 m/s burn at the low point adds 1.03 MJ for every kilogram; the same burn at the high point, where the spacecraft moves at only 1.57 km/s, adds 0.16 MJ. Same fuel, more than six times the energy.

Oberth effectThe name engineers give to this extra gain from firing an engine where a craft moves fastest.

So the burns that stretched the orbit fired at the low point, and each one pushed the high point further out. One burn did the opposite job: to lift the low point, you fire at the high point, the same rule turned round. Press Next below to go through them one at a time; the spacecraft flies each orbit at its true relative speeds, racing past Earth at the lowest point and crawling at the highest.

Chandrayaan-3's orbits around Earth, to scale Two panels, both to scale. Top: Earth with the spacecraft's six orbits around it, each a long ellipse. Every orbit's lowest point is only about 200 kilometres up, so it seems to touch Earth on the left; the burns that enlarged the orbit were made there. The highest points, to the right, are 36,500, 41,762, 41,603, 51,400, 71,351 and 127,609 kilometres above the surface, marked on a height scale that starts at Earth's surface. Bottom: the whole trip, at a smaller scale, with the same orbits now tiny around Earth and the path to the Moon stretching about 370,000 kilometres to the right, where it meets the Moon's orbit. Around Earth, to scale: the orbit after launch and after each burn lowest point (perigee), about 200 km up: too close to Earth to see at this scale. The burns that enlarged the orbit were made here. 0 36,500 41,762 51,400 71,351 127,609 Dashed lines: each orbit's highest point (apogee), in km above Earth's surface The whole trip, to scale close-up above Earth path to the Moon from 1 August (simplified) Moon (drawn ×3 size) Moon's orbit, 384,400 km from Earth Chandrayaan-3's orbits around Earth, to scale Two panels, both to scale. Top: Earth with the spacecraft's six orbits around it, each a long ellipse. Every orbit's lowest point is only about 200 kilometres up, so it seems to touch Earth on the left; the burns that enlarged the orbit were made there. The highest points, to the right, are 36,500, 41,762, 41,603, 51,400, 71,351 and 127,609 kilometres above the surface, marked on a height scale that starts at Earth's surface. Bottom: the whole trip, at a smaller scale, with the same orbits now tiny around Earth and the path to the Moon stretching about 370,000 kilometres to the right, where it meets the Moon's orbit. Around Earth, to scale lowest point (perigee), ~200 km up 0 36,500 51,400 71,351 127,609 highest points (apogee), km above surface The whole trip, to scale close-up above Earth path to the Moon (simplified) Moon (drawn ×4 size) Moon's orbit
  1. 14 July · Launch. LVM3 leaves the 3,895 kg spacecraft in an orbit that comes within 170 km of Earth's surface at its lowest point and goes out to 36,500 km at its highest. One trip around takes 10.7 hours; at the lowest point it moves at 10.28 km/s.
  2. 15 July · Burn 1. New orbit: lowest point 173 km, highest point 41,762 km above Earth. One trip around takes 12.5 hours. Speed at the lowest point: 10.35 km/s.
  3. 17 July · Burn 2. This burn lifted the lowest point, from 173 km to 226 km; the highest point stayed about the same (41,603 km). With the lowest point higher, the speed there is slightly lower: 10.30 km/s.
  4. 18 July · Burn 3. New orbit: lowest point 228 km, highest point 51,400 km above Earth. One trip around takes 16.0 hours. Speed at the lowest point: 10.41 km/s.
  5. 20 July · Burn 4. New orbit: lowest point 233 km, highest point 71,351 km above Earth. One trip around takes 23.9 hours. Speed at the lowest point: 10.54 km/s.
  6. 25 July · Burn 5. New orbit: lowest point 236 km, highest point 127,609 km above Earth. One trip around takes 51.5 hours. Speed at the lowest point: 10.72 km/s.
  7. 1 August · Sent towards the Moon. New orbit: lowest point 288 km, highest point 369,328 km above Earth, out at the Moon's distance. Speed at the lowest point: 10.84 km/s, 99% of the speed needed to escape Earth from there (10.94 km/s). The Moon's gravity caught it on 5 August.
Climbing away from Earth, to scale. Each orbit is ISRO's published figure, drawn to scale; the numbers in the steps are worked out from them. The orbits are drawn on one shared axis so that they can be compared; the real ones did not line up exactly.

The speed at the low point follows from two conservation laws. Between the low point (distance \( r_p \), speed \( v_p \)) and the high point (\( r_a \), \( v_a \)), energy and angular momentum are both conserved:

$$ \tfrac12 v_p^2 - \frac{GM}{r_p} = \tfrac12 v_a^2 - \frac{GM}{r_a}, \qquad r_p v_p = r_a v_a $$

In words: kinetic plus potential energy (per kilogram) is the same at both ends of the orbit; and distance times speed is the same at both ends, so the spacecraft is 6.5 times slower at the highest point of the launch orbit, which is 6.5 times as far from Earth's centre.

Take \( v_a = v_p r_p / r_a \) from the second equation, put it into the first, and solve for \( v_p \):

$$ v_p = \sqrt{\frac{2GM\,r_a}{r_p\,(r_p + r_a)}} $$

Here is the whole calculation for the launch orbit, from ISRO's two heights to the speed, the time for one orbit and the energy.

Worked example: the launch orbit, lowest point 170 km and highest point 36,500 km above Earth

Given: \( GM = 3.986 \times 10^{14}\ \mathrm{m^3/s^2} \) for Earth, and Earth's radius \( R = 6{,}378\ \mathrm{km} \). Work in metres and seconds inside the square roots.

  1. Lowest point, measured from Earth's centre\( r_p = 6{,}378 + 170 = 6{,}548\ \mathrm{km} \)
  2. Highest point, measured from Earth's centre\( r_a = 6{,}378 + 36{,}500 = 42{,}878\ \mathrm{km} \)
  3. The orbit's size, its semi-major axis\( a = \dfrac{r_p + r_a}{2} = 24{,}713\ \mathrm{km} \)
  4. Speed at the lowest point\( v_p = \sqrt{\dfrac{2GM\,r_a}{r_p\,(r_p + r_a)}} = 10.28\ \mathrm{km/s} \)
  5. Speed at the highest point (angular momentum: r × v is the same at both)\( v_a = v_p \times \dfrac{r_p}{r_a} = 1.57\ \mathrm{km/s} \)
  6. Time for one trip around\( T = 2\pi\sqrt{\dfrac{a^3}{GM}} = 38{,}664\ \mathrm{s} \approx 10.7\ \text{hours} \)
  7. Energy per kilogram (negative: still held by Earth)\( \dfrac{E}{m} = -\dfrac{GM}{2a} = -8.06\ \mathrm{MJ/kg} \)

Every row of the table below is these same seven steps, with that row's two heights.

ISRO's orbits, and what follows from them. The two heights are ISRO's; the last three columns are worked out from them, as in the example above.
DateStepLowest point, km upHighest point, km upOne trip aroundSpeed at lowest point, km/sEnergy per kg, MJ
14 JulLaunch (LVM3)17036,50010.7 hours10.28−8.06
15 JulBurn 117341,76212.5 hours10.35−7.29
17 JulBurn 222641,60312.5 hours10.30−7.30
18 JulBurn 322851,40016.0 hours10.41−6.19
20 JulBurn 423371,35123.9 hours10.54−4.73
25 JulBurn 5236127,60951.5 hours10.72−2.83
1 AugSent towards the Moon288369,328—10.84−1.04
Around the Moon: heights above the Moon, speeds and times relative to the Moon
5 AugCaptured16418,07428.2 hours2.17—
6 AugOrbit lowered1704,3136.3 hours1.98—
9 AugOrbit lowered1741,4373.2 hours1.79—
14 AugOrbit lowered1501772.1 hours1.62—
16 AugOrbit lowered1531632.1 hours1.61—
18 AugLander, its own first burn1131572.0 hours1.64—
20 AugLander, final orbit251341.9 hours1.69—

The last burn around Earth, early on 1 August, pushed the speed at the lowest point to 10.84 km/s: 99% of escape speed there. That was enough to coast out to the Moon's distance, losing speed all the way, as the energy chart shows.

Orbital energy per kilogram after each burn A bar chart of the spacecraft's orbital energy per kilogram after each step. Every value is below zero, because a body in orbit is still held by Earth; zero would mean just enough energy to escape. The deeper the bar, the more tightly Earth holds it. Launch, minus 8.06 megajoules per kilogram; burn 1, minus 7.29; burn 2, minus 7.30, almost unchanged because that burn lifted the lowest point instead; burn 3, minus 6.19; burn 4, minus 4.73; burn 5, minus 2.83; and the burn that sent it to the Moon, minus 1.04. Orbital energy per kg of spacecraft (MJ = megajoules) 0 escape −2 −4 −6 −8 Launch −8.06 Burn 1 −7.29 Burn 2 −7.30 Burn 3 −6.19 Burn 4 −4.73 Burn 5 −2.83 To Moon −1.04 burn 2 lifted thelowest point instead deeper bar =held more tightlyby Earth Orbital energy per kilogram after each burn A bar chart of the spacecraft's orbital energy per kilogram after each step. Every value is below zero, because a body in orbit is still held by Earth; zero would mean just enough energy to escape. The deeper the bar, the more tightly Earth holds it. Launch, minus 8.06 megajoules per kilogram; burn 1, minus 7.29; burn 2, minus 7.30, almost unchanged because that burn lifted the lowest point instead; burn 3, minus 6.19; burn 4, minus 4.73; burn 5, minus 2.83; and the burn that sent it to the Moon, minus 1.04. Energy per kg (MJ = megajoules) 0 = escape Launch · −8.06 Burn 1 · −7.29 Burn 2 · −7.30 Burn 3 · −6.19 Burn 4 · −4.73 Burn 5 · −2.83 To the Moon · −1.04 Longer bar = held more tightly by Earth.Burn 2 lifted the lowest point instead.
Energy per kilogram, \( -GM/2a \), after each step. All the values are negative because the spacecraft is still held by Earth; zero would mean escaping for good. Each burn made the number less negative, from −8.06 to −1.04 MJ/kg; the Moon's gravity did the rest.

Why did each step take days?

A one-day orbitAfter burn 4, the orbit's size \( a \) was 42,170 km, almost exactly that of a geostationary orbit, 42,164 km: the orbit in which a satellite goes round once a day and so stays over one spot on the equator. So one trip around took almost exactly a day.

Because a burn that enlarges the orbit has to happen at the lowest point, and the spacecraft has to come all the way back round to it. Kepler's third law gives the time for one trip around, the period \( T \):

$$ T = 2\pi\sqrt{\frac{a^3}{GM}} $$

In words: the period depends only on the orbit's size. Make \( a \) twice as big and one trip around takes \( 2^{3/2} \approx 2.8 \) times as long.

On the launch orbit, one trip around took 10.7 hours. After burn 4 it took 23.9 hours; after burn 5, 51.5 hours, over two days between chances to fire. ISRO also measured each new orbit before planning the next burn. And the last burn had to wait for the Moon, because the spacecraft had to arrive where the Moon would be about four days later.

One burn did something different. On 17 July the second burn lifted the lowest point, from 173 km to 226 km above Earth, and barely changed the energy. You can see it in the chart: burn 2 is the bar that does not move.

How did it slow down at the Moon?

By firing backwards. On 5 August, at its closest approach, the propulsion module fired against the direction of motion for 1,835 seconds. Afterwards it was moving at 2.17 km/s at its lowest point, under the 2.27 km/s it would have needed to escape the Moon from there, and the Moon had it: an orbit 164 km above the Moon at its lowest point and 18,074 km at its highest.

Then the climb ran in reverse. Four more burns against the motion shrank the orbit until, by 16 August, it was nearly circular: 153 km above the Moon at its lowest point and 163 km at its highest. The next day the lander separated, and on 18 and 20 August it lowered its own orbit to 25 km at the lowest point and 134 km at the highest, the orbit it would land from.

Chandrayaan-3's orbits around the Moon Two panels. Top, to scale: the Moon with the spacecraft's orbits around it. The capture orbit's highest point is 18,074 kilometres above the Moon; later orbits reach 4,313 and then 1,437 kilometres, and the last ones are nearly circular, 150 to 180 kilometres up, hugging the Moon. Every lowest point is about 150 to 175 kilometres up. The landing site is marked near 69 degrees south. Bottom: the last orbits as height above the Moon over one trip around, lowest points lined up: 150 to 177, 153 to 163, 113 to 157 and finally 25 to 134 kilometres, the orbit the lander came down from. Around the Moon, to scale landing site, 69.4°S 18,074 km 4,313 km 1,437 km Labels give each orbit's highest point above the Moon; every lowest point is 150–175 km up. The last orbits: height above the Moon during one trip around Moon's surface · left to right: one trip around, with each orbit's lowest point in the middle 0 50 100 150 height, km 14 Aug: 150 to 177 km 16 Aug: 153 to 163 km 18 Aug: 113 to 157 km 20 Aug: 25 to 134 km descent and landing Chandrayaan-3's orbits around the Moon Two panels. Top, to scale: the Moon with the spacecraft's orbits around it. The capture orbit's highest point is 18,074 kilometres above the Moon; later orbits reach 4,313 and then 1,437 kilometres, and the last ones are nearly circular, 150 to 180 kilometres up, hugging the Moon. Every lowest point is about 150 to 175 kilometres up. The landing site is marked near 69 degrees south. Bottom: the last orbits as height above the Moon over one trip around, lowest points lined up: 150 to 177, 153 to 163, 113 to 157 and finally 25 to 134 kilometres, the orbit the lander came down from. Around the Moon, to scale landing site, 69.4°S 18,074 km 4,313 km 1,437 km labels: highest point above the Moon The last orbits: height over one orbit one trip around, lowest point in the middle 0 50 100 150 height, km 14 Aug: 150 to 177 km 16 Aug: 153 to 163 km 18 Aug: 113 to 157 km 20 Aug: 25 to 134 km descent and landing
  1. 5 August · Captured by the Moon. At its closest approach the engine fired against the direction of travel for 1,835 seconds (about 31 minutes), slowing it enough for the Moon to hold it. Orbit: lowest point 164 km, highest point 18,074 km above the Moon. One trip around takes 28.2 hours.
  2. 6 August · Lowered. Lowest point 170 km, highest point 4,313 km above the Moon. One trip around takes 6.3 hours.
  3. 9 August · Lowered. Lowest point 174 km, highest point 1,437 km above the Moon. One trip around takes 3.2 hours.
  4. 14 August · Nearly circular. Lowest point 150 km, highest point 177 km above the Moon. One trip around takes 2.1 hours.
  5. 16 August · Circular. Lowest point 153 km, highest point 163 km above the Moon. One trip around takes 2.1 hours. Next day the lander separated from the propulsion module.
  6. 18 August · The lander's own first burn. Lowest point 113 km, highest point 157 km above the Moon.
  7. 20 August · The lander's final orbit. Lowest point 25 km, highest point 134 km: the orbit it landed from.
  8. 23 August · Landing. Just after 6 pm IST the lander fires its engines from the low side of its orbit and comes down near 69.4°S, 32.3°E.
Settling down to the Moon. Top, to scale: the last orbits sit so close to the surface that they are drawn as a height profile below, where the vertical axis is stretched.

Why land on 23 August?

The landing siteNear 69.4°S, 32.3°E, about 600 km from the lunar south pole, since named Shiv Shakti point.

Because of sunrise. Vikram and the Pragyan rover ran on solar power and were built to work for one lunar day, about 14 Earth days. ISRO set the landing for the start of daylight at the site, so that the instruments would have the whole day.

So the forty days were not slowness. ISRO's earlier missions, Chandrayaan-1, Chandrayaan-2 and the Mars Orbiter Mission, raised their orbits the same way. A smaller rocket and a patient spacecraft can reach the same place as a giant rocket, if you spend time instead of fuel. The rules that decide the trade are the ones in your gravitation chapter.

Try it: JEE practice

Take \( GM = 3.986 \times 10^{14}\ \mathrm{m^{3}\,s^{-2}} \) for Earth and Earth's radius as 6,378 km.

  1. After its fifth burn, ISRO reported an orbit of 236 km × 127,609 km: lowest point 236 km and highest point 127,609 km above Earth. (a) Using conservation of energy and angular momentum, find the speed at the lowest point. (b) Find the time for one trip around.
  2. The launch orbit's lowest and highest points were 170 km and 36,500 km above Earth; the orbit that took it to the Moon, 288 km and 369,328 km. How much energy per kilogram did the spacecraft gain between them?
  3. A spacecraft can make the same small burn \( \Delta v \) at the lowest point or the highest point of its orbit. Show that the energy gained is larger at the low point by the factor \( r_a / r_p \), and find it for the launch orbit.
  4. On the launch orbit, one trip around took 10.7 hours. Using only Kepler's third law, find the time for one trip around after burn 4 (lowest point 233 km, highest point 71,351 km above Earth).
Answers

1. Add Earth's radius to each height: \( r_p = 6.614 \times 10^{6}\ \mathrm{m} \) and \( r_a = 1.340 \times 10^{8}\ \mathrm{m} \). (a) \( v_p = \sqrt{2GM\,r_a / (r_p (r_p + r_a))} \approx 10.72\ \mathrm{km\,s^{-1}} \). (b) \( a = (r_p + r_a)/2 = 7.030 \times 10^{7}\ \mathrm{m} \), so \( T = 2\pi\sqrt{a^{3}/GM} \approx 1.86 \times 10^{5}\ \mathrm{s} \), about 51.5 hours.

2. \( E/m = -GM/2a \): −8.06 MJ/kg for the launch orbit (\( a = 2.471 \times 10^{7}\ \mathrm{m} \)) and −1.04 MJ/kg for the orbit that took it to the Moon (\( a = 1.912 \times 10^{8}\ \mathrm{m} \)): a gain of about 7.0 MJ for every kilogram.

3. For a small burn, \( \Delta E \approx m\,v\,\Delta v \), so the ratio is \( v_p / v_a \), which angular momentum makes equal to \( r_a / r_p \). For the launch orbit, \( r_a / r_p = 42{,}878 / 6{,}548 \approx 6.5 \).

4. \( T \propto a^{3/2} \). The semi-major axis grew from 24,713 km to 42,170 km, so \( T = 10.7 \times (42{,}170 / 24{,}713)^{3/2} \approx 23.9 \) hours: almost exactly a day.

Read: Gravitation notes · Practise: JEE chapter-wise tests

Sources

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