(Layman Science Series) Week 3 - When is hottest day or days in Jamaica?

[crosswire]

Intro
I am trying to post a science/physics question of varying difficulty each week for this year.

The Rules are:
Answers must be complete, and each statement must be sound/logical.
The better the answer, then the more points awarded. Answers can always be updated at anytime as
we learn something new everyday.
The greater the difficulty, the greater the points. I will give an answer at end of week. Repeating my
answer awards no points, you will have to come with a better or alternate answer.

Given:
https://en.wikipedia.org/wiki/Tropic_of_Cancer

The Tropic of Cancer, which is also referred to as the Northern Tropic, is the most northerly circle of latitude on Earth at which the Sun can be directly overhead. This occurs on the June solstice, when the Northern Hemisphere is tilted toward the Sun to its maximum extent. It also reaches 90 degrees below the horizon at solar midnight on the December Solstice. Using a continuously updated formula, the circle is currently 23°26′11.1″ (or 23.43642°) north of the Equator.

https://en.wikipedia.org/wiki/Summer_solstice

The summer solstice, also known as estival solstice or midsummer, occurs when one of Earth's poles has its maximum tilt toward the Sun. It happens twice yearly, once in each hemisphere (Northern and Southern). For that hemisphere, the summer solstice is when the Sun reaches its highest position in the sky (for areas outside of the tropics) and is the day with the longest period of daylight. Within the Arctic circle (for the northern hemisphere) or Antarctic circle (for the southern hemisphere), there is continuous daylight around the summer solstice. On the summer solstice, Earth's maximum axial tilt toward the Sun is 23.44°. Likewise, the Sun's declination from the celestial equator is 23.44°.

The summer solstice occurs during summer. This is the June solstice in the Northern Hemisphere and the December solstice in the Southern Hemisphere. Depending on the shift of the calendar, the summer solstice occurs sometime between June 20 and June 22 in the Northern Hemisphere and between December 20 and December 23 in the Southern Hemisphere. The same dates in the opposite hemisphere are referred to as the winter solstice.

Earth's Equatorial Radius = 6378km
Earth's Polar Radius = 6357km
Latitude of Jamaica = 18.1° N
Latitude of the Great Pyramid of Gaza = 29.9792° N

Define a hottest day of a place as: the day in which at noon the sun is at its most center in sky.

You may assume that the orbit of the earth is circular for ease of calculations. (The orbit is not perfectly circular)

Question

a) How many hottest day(s) does Jamaica have each year?
b) When is/are the hottest day(s) of Jamaica within ∓ 3 days [Show explanation or calculation]?
(2 Points)
Hint 1: Not all information given is needed
Hint 2: This is an approximation question. Maths is not intended to be hard.

https://worldmapwithcountries.net/20...r-and-tropics/



Difficulty
A-Level Geography
A-Level Mathematics

Just a side note showing of how much of Africa is in the tropics i.e Cancer to Capricorn.
https://en.wikipedia.org/wiki/Africa...rojection).svg




Site share of the week
StarTalk - YouTube
https://www.youtube.com/channel/UCqo...PjEUFcF2FklnBA

[crosswire]

Learnt something this morning so I have to make this amendment.

Define a hottest day of a place as: the day in which at noon that place is closest to the sun.
Define a hottest day of a place as: the day in which at noon the sun is at its most center in sky.

https://www.space.com/3304-earth-clo...ad-winter.html

Earth's orbit is not a perfect circle. It is elliptical, or slightly oval-shaped. This means there is one point in the orbit where Earth is closest to the Sun, and another where Earth is farthest from the Sun. The closest point occurs in early January, and the far point happens in early July (July 7, 2007).

If you plot the noontime position of the Sun in the sky over a one-year period, it produces a figure-eight shape on the sky (Figure A). This is the analemma. You may have seen it drawn on a globe of Earth. The shape results from the combination of two things: the 23.5° tilt of Earth on its spin axis, and the elliptical shape of Earth's orbit around the Sun.

The orbital distance is not really what causes the seasons as explained on space.com. I realize that it must be something different, and now take the cause to be how much the sun's incident ray are at 90° to the earth's surface, i.e. the degree to which the sun is more centered in the sky at noon.

You may assume that the orbit of the earth is circular for ease of calculations. (The orbit is not perfectly circular)

[crosswire]

Hints: The sun's rays is always perpendicular to the earth's surface at some point. That point lies on a "latitude" value. That latitude value is between the Tropic of Cancer and the Tropic of Capricorn. The earth's orbit is assumed circular (revolution). The tilt and revolution causes this "latitude" value to vary.

Mathematics is technically not a science subject, however it is a tool (just like English) that is needed by science to complete science, or else science would not be what it is now. Good luck figuring out the maths.

[khat17]

Hi. I'd like to propose that it's not necessarily the "hottest" in Tropical areas like Jamaica. The greenhouse effect still occurs even without the excessive pollution. And it's possible to get sunburn on a snowy day. So that's my input for this one thus far. Hopefully it makes persons think.

Hints: The sun's rays is always perpendicular to the earth's surface at some point. That point lies on a "latitude" value. That latitude value is between the Tropic of Cancer and the Tropic of Capricorn. The earth's orbit is assumed circular (revolution). The tilt and revolution causes this "latitude" value to vary.

Mathematics is technically not a science subject, however it is a tool (just like English) that is needed by science to complete science, or else science would not be what it is now. Good luck figuring out the maths.

There's also the fact that the Sun is far larger than the Earth. So why is it "hotter" in the middle than at the extremities?

[crosswire]

Hi. I'd like to propose that it's not necessarily the "hottest" in Tropical areas like Jamaica. The greenhouse effect still occurs even without the excessive pollution. And it's possible to get sunburn on a snowy day. So that's my input for this one thus far. Hopefully it makes persons think.

There's also the fact that the Sun is far larger than the Earth. So why is it "hotter" in the middle than at the extremities?

Well there will be effects of climate change and pollution (green house) that would change the temperature each day for a specific place, even Jamaica. But for this question you can assume that the hottest day a specific place can be for the year is when the midday 12noon sun is at the most center in the sky for the entire year. Ignore cloudy weather, and tree coverage, and proximity to breezy beaches. Generally over many years a climate pattern is formed.

Thus I can rephrase the question to which day(s) in Jamaica is the sun most in the center of the sky at noon. (Unfortunately this makes it easier for people to google the answer. However, the working out is required to get the two marks)

Also unfortunately, providing and writing the answers is a bit more time consuming than posting the question.

More Hints: In Geometry, components of 3d space can be treated independently.
You can choose suitable reference points, and then use the perspective from that view.

[crosswire]

You are expected to know that the earth's orbit around the sun is approximately 365 days (Gregorian Calendar)

P.S. I recently found out the our calendar is based on the solar system where the axis of earth experience precession. As a result the stars orientation varies at each solstice. This is the Gregorian calendar.

Rant:
Consider this math problem: Alice has twice as much apples as Bob, then gives Eve 2 and now has 2 more than Bob. How much does Bob have?

Best way to solve is to use simultaneous equations. But what if you do not know simultaneous equation, you could use brut force and just try whole integers 1, 2, 3, 4

In your head, assume 1 for Bob.... You find that 1 that does not work. 1 is way too small.
In your head, assume 2 for Bob.... You find that 2 still does not work. 2 is still too small.
In your head, assume 3 for Bob.... You find that 3 almost works. 3 is just a bit small.
In your head, assume 4 for Bob.... You find that 4 works.
In your head, assume 5 for Bob.... You find that 5 is now too big. An apparent trend emerges.

Generally when doing algebra, you should work the problem in your head in a rough way so as to check if your calculations can pass some basic validation test.


Solution:
Assume a circular orbit. The sun's rays is always perpendicular to a normal plan of the earth's surface, with some surface intercept point x.
Ignore daily rotation of earth, but focus on the tilted axis and corresponding latitudes. Picture a revolution about the sun.

Picture the earth in initial position A, and north tilted towards the sun. This initial is the summer solstice (for northern hemisphere), x is on the Tropic of Cancer.

In position B, 1/4 revolution from A, x is on the equator

In position C, 1/2 revolution from A, x is on the Tropic of Capricorn. Picture the earth with north pole titled maximum away from sun.

In position D, 3/4 revolution from A, x is on the equator.

As a matter of perspective, earth going around sun is similar to if earth was still and sun went around the earth. Picture that "orbital plane" of the sun and project it on to the surface of the earth.
It intercepts the earths surface and forms a circle X with points x (variable).
Because the earth is tilted, the equator would be a circle E with its plane tilted to the plane of circle X

Remember X is formed from the yearly orbit.

At 0 degree tilt, X and E would correspond.

From the perspective of E, X is a variation in the latitude from E (as you vary x).

At 90 degree tilt, X would show that the variation of the latitude would go all the way to the north pole and to the south pole within a revolution. The latitude variation, as x traverse circle X would be a sinusoidal variation.

At 23.44 degree tilt, the variation is still sinusoidal but the amplitude is less (At 0 degrees title the amplitude is 0) The amplitude of the sinusoid varies from the Tropic of Cancer to the Tropic of Capricorn

Answer to part 1, Jamaica would have two days when x is on the latitude of Jamaica
See diagram:


Note: Places outside of the tropics should not experience a 90 degree incident at any time.

Answer to part 2:

Normalizing x in unit of (The tropic of Cancer)
On the summer solstice, Jun 21, position A, x has the value of a = 1 * (Tropic of Cancer) correspond to cosine (0)

For Jamaica at 18.1 ± 0.2 degree latitude,
On 1st hot day, position H1, x = 18.1/23.4 = cos(hot1) ==> hot1 = arcos(18.1/23.4) = ±39.33 degrees of orbit = ±39.33/360 * 356 days = 39 days before summer solstice ==> hot2 = 39 days after summer solstice
hot1 ~ May 13
hot2 ~ July 30

July 30 expected to be warmer because it follows

2B=A
B+2=A-2
B=A-4

B=2B-4
B=4

[khat17]

Nice one. I'm not a "math person" but I appreciate it. I'm well aware that many things use mathematical formulas to get certain absolutes - my issue would be the formulas.

I apologize for not being around more. Was busy with work. If nobody else enjoys the science week, I do. So please keep it up. It also provides a platform where non-members can look on and work their brain cells a bit.

[crosswire]

I apologize for not being around more. Was busy with work. If nobody else enjoys the science week, I do. So please keep it up. It also provides a platform where non-members can look on and work their brain cells a bit.

Thanks Khat. Will try.

Yes we all have different forms of stress to deal with at times, and for ourselves, we need to push through.