Intuition The ONE core ideaA satellite around Earth is also pulled by the Moon and Sun — but what actually disturbs its orbit is not their raw pull, it is how much more (or less) they pull the satellite than they pull the Earth itself. Because we watch the satellite from an Earth that is itself falling toward the Moon, only the tiny difference across the Earth-to-satellite gap survives — and that difference is what slowly warps the orbit.
This page builds every single symbol and idea the parent note leans on, starting from a smart 12-year-old who has never seen an arrow with a hat on it. Read top to bottom; each block earns the next.
Definition Vector and the notationr vector is an arrow: it has a length (how far) and a direction (which way). We write it with a small arrow on top, like r length , we drop the arrow and write r
Look at the first figure. The blue arrow starts at a point and ends at another. That whole arrow is r r
Intuition Why the topic needs arrows at allThe Moon does not just pull "a bit" — it pulls in a particular direction (toward the Moon), and how much depends on where the satellite sits. Only arrows can carry both facts at once. A plain number could never tell us the pull tilts sideways when the satellite is off to the side.
Position an arrow
r from a chosen origin to where a thing is
The bare symbol r the
length of
r , a single number
Every arrow has to start somewhere . That starting point is the origin .
r s r E r 3 inertial origin — one that is not accelerating). From it draw three arrows:
r s satellite ,r E Earth ,r 3 third body (Moon or Sun). The little "3" just means "the third mass in the story."
Definition Relative position — subtracting arrowsr = r s − r E from Earth to the satellite . Subtracting one position arrow from another gives you "where is A as seen from B." Likewise d = r 3 − r E from Earth to the third body .
Look at the second figure: the three long arrows come out of the far origin, and the short green arrow r r s r E
Intuition Why "relative" matters hereWe never track the satellite from that far-away origin — we track it from Earth , because Earth is our home base. That is the whole reason the perturbation comes out as a difference : our home base is moving, so we must always ask "relative to Earth." This is the Two-body problem mindset: reduce two moving bodies to one relative arrow r
r satellite position
relative to Earth ,
r s − r E d third body position relative to Earth,
r 3 − r E Why we subtract r E because we observe from Earth, not from the inertial origin
Now the pull itself. We break the famous law into bite-sized symbols.
G M μ = GM
G gravitational constant , a fixed number of nature (6.674 × 1 0 − 11 strength of gravity for the whole universe.M mass of the pulling body (kilograms).Their product μ = GM standard gravitational parameter . We bundle them because in orbit maths G M always travel together — so we give the pair one name. For Earth, μ = G M E μ 3 = G m 3
Definition The unit arrowr ^ hat means "length exactly 1, direction only." So r ^ = r / r r direction and throws away the size . Use it whenever you want to point somewhere without saying how far.
Look at the third figure to see why r 3 r r 2 r ^ r r r r
Common mistake "The hat and the arrow are the same."Why it feels right: both point the same way. Why wrong: r r r ^ Fix: r ^ = r / r direction-only version.
Intuition Why we need "acceleration," not "force"Every object at the same spot gets the same acceleration from a given mass, regardless of its own weight (heavy and light fall together). Since we compare how the Moon accelerates the satellite versus the Earth , working in accelerations lets the satellite's own mass drop out entirely — that is the secret behind the cancellation to come.
μ the bundle GM
r ^ unit vector — direction of
r , length exactly 1
The minus sign in − μ r / r 3 gravity pulls inward , toward the mass
Why r 3 r / r 3 it is
r ^ / r 2 rewritten; one
r converts
r into
r ^
r ¨ r ˙ Two dots, r ¨ acceleration , how fast the velocity changes. This is a derivative taken twice with respect to time.
Intuition Why the topic needsr ¨ acceleration : force sets how the velocity changes. So the whole derivation is a sentence of the form "r ¨ =
r ˙ velocity — how position changes each second
r ¨ acceleration — how velocity changes each second
The tidal formula uses d ^ ⋅ r r along the direction to the third body?"
a ⋅ b dot product of two arrows is a single number: multiply their lengths and then by how aligned they are. In particular, d ^ ⋅ r d ^ length of the shadow of r d ^ r
Look at the fourth figure: drop a straight-down shadow of the green arrow r d ^ d ^ ⋅ r r d ^ r
Intuition Why the topic needs the dot productThe tidal effect stretches along the line to the Moon and squeezes sideways . To write that mathematically you must split r d ^ 3 ( d ^ ⋅ r ) d ^ d ^
d ^ ⋅ r the shadow of
r onto the direction
d ^ (its "along-the-Moon" part)
Dot product of perpendicular arrows zero
Why the tidal formula uses it to separate the stretch direction from the squeeze direction
Now every piece is in hand, so we can name the central object.
s = r 3 − r s s from the satellite to the third body — how the Moon "sees" the satellite. The parent note's key quantity is not one pull but a difference : the Moon's pull on the satellite minus its pull on the Earth. Because Earth is free-falling toward the Moon along with the satellite, the shared part of the pull cancels, and only the variation across the gap survives. That leftover is the tidal (differential) acceleration.
Intuition Why "difference," picturedImagine two people falling in the same lift. Neither feels the fall — only if one is pulled harder than the other do they drift apart. The satellite and Earth are the two fallers; the Moon pulls them very slightly differently, and that tiny drift is the perturbation. This is the same free-fall logic that makes astronauts float.
s satellite-to-third-body arrow,
r 3 − r s Why the raw pull μ 3 / s 2 not the perturbation Earth is pulled almost equally; the common part cancels, only the difference remains
The surviving difference is called the tidal / differential acceleration
≪ r / d r ≪ d r much smaller than d r d r / d tiny number . Whenever a quantity is tiny, we may throw away its square (tiny² is even tinier) — that is why the parent expands "to first order in r / d
Intuition Why approximate at allThe exact difference-of-two-pulls formula is correct but clumsy. Because r / d tidal form d 3 μ 3 ( 3 ( d ^ ⋅ r ) d ^ − r )
r ≪ d Earth–satellite distance is far smaller than Earth–third-body distance
Why we drop ( r / d ) 2 it is negligibly small compared to the kept first-order term
Vector arrow r-hat and length r
Relative position r equals r_s minus r_E
G and mass give mu equals GM
Newton pull minus mu r over r cubed
Double dot r means acceleration
Split along versus perpendicular
Tidal approximation with r much less than d
Test yourself — you are ready for the parent note only if each reveal feels obvious.
I can say what the arrow and hat mean r is a full arrow (direction + length
r );
r ^ = r / r is direction only, length 1
I can build r r s r E r = r s − r E , the arrow from Earth to the satellite
I know what μ μ = GM μ 3 = G m 3
I can read Newton's pull a = − μ r / r 3 = − μ r ^ / r 2 , pointing inward, weakening as
1/ r 2 I know what r ¨ acceleration, the second time-derivative of position
I can interpret d ^ ⋅ r the shadow (projection) of
r along the direction to the third body
I understand why the perturbation is a difference Earth free-falls too; the common pull cancels, only the tidal gradient remains
I know why we expand in r / d r ≪ d r / d
