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Sidereal Day vs Solar Day

okay now we're going to discuss the

difference between the sidereal day and

the solar day okay so what are their

differences so the one that we say one

day is 24 hours one day is 24 hours

which of this is it one day is 24 hours

it's the solar day it's the time that it

takes for the Sun to start at a certain

position in the sky for it to go around

come back to the same position in the

sky right that's the solar day that one

is 24 hours so I want to know sidereal

day means if a certain star is at a

certain location in the sky how many

hours does it take for the star to go

around come back to the same location as

it turns out it's gonna be a little less

than 24 hours let's see why okay so I'm

gonna draw the Sun here I'm gonna draw

the earth I'm gonna say let this

represent a certain point and that point

is facing the Sun right and so now as

the earth this is a top view as the

earth is going around the Sun right

revolution the earth is also rotating so

in this lecture I'm going to focus on

revolution rotation okay so the earth is

rotating like this like that and the

earth is revolving like that so by the

time the earth has gone around one day

because it's somewhere here for

exaggeration purposes I'm making it kind

of big difference right this goes around

it goes around because around so it goes

around

finally one solar day will be

represented by the time that it takes

for that dot to be in line right from

the Sun here and if I draw a line from

the Sun over there that's one solar day

that point is facing the Sun that point

is also facing the Sun that's 24 hours

right now what's sidereal day in this

picture what would the sidereal day be

so sidereal day

we'll be the time that it takes for that

point the point was facing this rate

right and then if the earth goes like

this and then this point is going like

that like that like that so the earth is

here the time that it takes for the

point to also be facing this way you see

so if you draw a line like that line

like that to the Stars right you draw a

line like that to the Sun to this to the

Stars like that right so this is what

sidereal day which one is longer the

sidereal day or the solar day well

because the rotation is like this right

it's going to go like that like this and

then a little bit longer the earth would

have to go tiny bit more for this stuff

to start going down when the doc goes

down it starts facing the Sun right so

this occurs before this occurs so in

order for me to illustrate this on top

of that what I can do is put a little

mark here and I can say when that dot

was over there a little bit earlier that

was the sidereal day right that means

the first was a little bit lower you see

that top was there right then as the

earth went up the dot rotated rotated so

that point the solar the sidereal day

point is left up here now okay so how

can we find this out so what we're gonna

do is we're gonna specify the location

of this dot from my XY coordinate what

are the locations of this dot okay so

that's gonna be my position vector R X

as a function of T Y as a function of T

right what's X X is the distance from

here to that dot okay and it's rotating

around so what is X going to be well X

is going to be the distance between the

Earth and the Sun times cosine of omega

revolution

tea right plus radius of the earth

cosine of Omega rotation T plus pi and

then and then I'll tell you what the y

component is so what what do we mean by

this well when T is equal to zero right

what's cosine of zero just 1 right so

that will be the distance of the Earth

to the Sun from the center of the Sun to

the center of the earth so that's this

one distance Earth Sun right so at the

beginning when T is 0 what's X distance

of Earth's Sun and if I put 0 here

what's gonna happen cosine 0 so this

part is your cosine of PI cosine of pi

is what negative 1 right so basically

you have when T equals to 0 X is equal

to what distance earth Sun and then this

is your cosine PI is negative 1 minus

radius of Earth okay that's good

so you see you got that point is at this

distance minus the radius of the earth

so that's location the X location of the

object ok then what's going to happen

over time to the X location the you're

gonna have because the earth is

revolving around the Sun right so it's

going this way think about what the

center of the earth is doing just think

about where the center of the earth is

doing through the point of time the

center of the earth is moving this way

right like that like that like that so

if I draw from here to the center of the

earth

what's the X component of that so what

this guy is doing is finding out where

the position of the center of the earth

is at any time you've got distance earth

Sun cosine of the rate of revolution of

Earth times T so you see this one is w

revolution t see that means how many

irradiance its covering in one year

right the rate of revolution and then

the time that has elapsed you're just

taking cosine of that that's going to

give you the X component of the center

of the earth

what is this second one doing then what

is doing is it's taking another axis

here XY axis right it's giving the

position of that point with respect to

the center of the earth

so you're adding the position of the

center of the earth with respect to the

Sun plus the point the position of the

point with respect to the center of the

earth you see so what is that radius of

Earth is the distance from here to here

cosine Omega rotation T plus pi its

initial angle was what PI right CC was

facing here then after that it's going

to rotate rotate rotate rotate rotate

rotate

okay what's the rate of rotation W

rotation which is what we're going to be

solving for okay so now you get the idea

of the logic of this you're getting the

distance of the center of the Earth from

the Sun plus the distance of that point

with respect to the center of the earth

now let's do the same thing to the white

component what's the white component

going to be the Y component is going to

be distance of Earth son sine of omega

revolution t plus the radius of Earth

sine Omega rotations Yi plus pi okay so

when ki is equal to zero what's the y

equal to okay

sine of zero would be zero sine of zero

plus Phi sine of PI would be zero okay

so both they're both zero yeah because

the point is over here initially so it's

not above the y-axis right as the center

of there

moving up above so the location of the

center of the earth would be sign of how

far it has revolved and then the

location of the point with respect to

the center of the earth would be sign of

how much it has rotated okay after one

solar day what's gonna happen where is

the point gonna be it's gonna be right

over here right so now I can do like a

picture like this after one solar day

and then call this angle theta so theta

tangent of theta is going to equal what

tangent of theta is going to be the Y

component which is this side divided by

the X component which is this one I know

what theta is yes I do because I

actually do know the rate of revolution

of the earth right that's one sidereal

year okay now what's the angle that it

has covered if it's it that's the rate

of revolution we have angle is equal to

W Rd times T so what is T what is one

solar day see this is one solar day but

what is this when we say that the

sidereal period of the earth is two

hundred and sixty five point two five

six three six three days what days are

we referring to those are also solar

days everything is in terms of solar

days right they when we say day it's

always a solar day right so that's 24

hours so all I do is just multiply this

number by one day okay

theta is equal to this thing

they cancel and therefore that will tell

me after one solar day how many really

the earth has revolved around the Sun

okay

so let's calculate that so what I'm

going to do is now this is a known value

right so I'm gonna erase this I'm going

to say tangent of 0.017 - or - one two

four one six two Rad's

is equal to distance of Earth from the

Sun so I'm going to put what's the

distance of the earth that's one a you

write one point four nine five now us

radius of Earth okay

radius of Earth is going to be six three

seven eight zero zero zero it's this six

thousand three hundred and seventy eight

kilometers so I'm changing that to

meters so that's gonna be six million

children and seventy eight million

meters then I'm gonna do what sine of

Omega rotation is unknown T is one just

once all of it so it's going to be Omega

rotation plus PI okay divided by this -

some word from Sun one point four nine

five nine okay let's just make sure

before we calculate anything that we put

everything in correctly this one was one

point four nine five nine seven eight

seven oh six this one was the angle from

before 101 72 old - one two four one six

- and then this one was 365 point two

five six three six three since the

numbers are very particular you have to

make sure you put it in all correctly

this one is missing one zero all all

four and then this is times one times

one and then this is the radius of the

earth okay so now let's calculate this

okay then you put this all into the

solver okay make sure you are in Radian

mode when you put this in sober so we'll

go over here so over now what kind of a

number am I expecting

the Omega rotation okay

well we know that if the earth starts

here in one solar day it's gonna go from

here

go all the way and then face towards the

Sun again right so this is the Sun so

what should be the rate of rotation now

notice that it's covering more than two

pi radians right in one day so it's

going or in going going coming and two

pi radians would have been that point

right two pi radians would have been the

point where he came it went all the way

around and then a little bit more so to

this equation I'm expecting an answer

something like Omega rotation greater

than two pi radians per day so I'm

expecting an answer greater than six

point two eight something something

something right so the Earth rotates in

such a fashion that in one solar day it

covers more than six point two eight

radians right so I'm going to go to my

answers now okay after looking over at

all the answers as I said there were

some negative answers there are some

positive answers by the answer that

we're interested in that physically

makes sense is something larger than six

point two eight so the answer comes out

to be six point millions per day so now

I can answer the question how long is

one sidereal day T sidereal day equals

two pi rads divided by Omega of rotation

equals two pi rads over this one C right

so when I just say this day goes up

right when I just say day and I don't

specify it that's the solar day then I

multiply this one day is 24 hours

multiply this by 24 express that in

hours and minutes you take point 9 3 49

3 multiplied by 60 so that would be 56

minutes 23 hours 56 minutes and then

subtract 56 from there multiplied by 60

the sidereal day is 23 hours 56 minutes

4 seconds so let's see what that means

if a certain star is at a certain point

right it will go around come back in 23

hours 56 minutes 4 seconds it's back to

where it is but if the Sun is at the

same place the Sun hasn't come fully

around yet right so the Sun will take 24

hours to go fooling around so it's the

sidereal day is shorter by about 4

minutes right from the solar day so

basically the Stars pull ahead of the

sun and then the sun starts falling

behind the stars ok so now you can see

the difference between the sidereal day

and the solar day this problem is very

interesting application of quite a lot

of trig as well so it shows you how you

can do some simple math some trig and

then analyze that put all of the data in

and be very careful about all the units

and all the sig figs and everything use

the t-i put it all and get the answer

and then so for some physical phenomenon

that we know in the universe ok so I

hope this helps you and you can do

further research on this if you want

thank you very much